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1 STATISTICS IN TRANSITION ew seres A Iteratoal Joural of the Polsh Statstcal Assocato CONTENTS Edtor s ote ad acowledgets... Subsso forato for authors... Saplg ad estato ethods BAGNATO L. PUNZO A. Noparaetrc bootstrap test for autoregressve addtve odels... PATEL J. PATEL P. A. O o-egatve ad proved varace estato for the rato estator uder the Mdzuo-Se saplg schee... ROSSA A. Estato of lfe-tables uder rght-cesorg... SHUKLA D. THAKUR N. S. PATHAK S. RAJPUT D. S.: Estato of ea uder putato of ssg data usg factor-type estator two-phase saplg... SRIVASTAVA M. K. SRIVASTAVA N. SINGH H. P. Full forato effcet estator of fte populato varace... ZIELIŃSKI W. A oparaetrc cofdece terval for at-rs-of-poverty-rate... Other artcles BUDN K. TATAR J. Kurtoss of a rado vector specal types of dstrbutos... CHATTERJEE S. UPADHAA L. N. SINGH J. B. NIGAM S. Cobed effect of fault detecto ad fault troducto rate o software relablty odelg... MUWANGA-ZAKE E.S.K. Motorg worers rettaces ad beefts Ugada: The Statstcal Issues... NEHREBECKA N. Teporal aspects of poverty Polad betwee by hazard odels... VERNIZZI A. Applyg the Hadaard product to decopose G cocetrato redstrbuto ad re-rag dexes... Boo revew Tas force o the qualty of the labour force survey. Fal report EUROSTAT Methodologes ad Worg papers 009 Edto 69 pages. Preparaed by J. Kordos... Reports The Deographc Future of Polad a scetfc coferece Łódź 7 8 Septeber VIII Coferece o Multvarate Statstcal Aalyss (MSA 009) Łódź Polad 6 8 Noveber Volue 0 Nuber Deceber 009

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3 STATISTICS IN TRANSITION-ew seres Deceber STATISTICS IN TRANSITION-ew seres Deceber 009 Vol. 0 No. pp EDITOR S NOTE AND ACKNOWLEDGEMENTS Sce ths ssue of the oural Statstcs Trasto ew seres s the last oe wth the past 009 year prtg cycle I would le to tae ths opportuty to express also o behalf of the Edtoral Offce our gratefuless to all the oural s patros ad supporters wth the Edtoral Board chared by Professor Józef Oleńs Presdet of the Cetral Statstcal Offce ad Assocate Edtors ad to tha all our collaborators authors ad referees as well as readers who otly cotrbuted to successful cotuato of the oural s sso. Altogether artcles by 65 authors were prted the oural durg the past year (copared to 7 artcles by 6 authors the prevous year). We would especally warly le to tha the people who acted as referees of the papers subtted for publcato durg the past year ther aes are lsted below followg the bref presetato of the cotets of ths ssue. *** Each of the fve artcles cluded frst part of ths ssue of the oural addresses soe d of the estato probles. I Noparaetrc Bootstrap Test for Autoregressve Addtve Models Luca Bagato ad Atoo Puzo as how to evaluate ad decde whether a addtve autoregressve odel of a type that are cooly used to descrbe ad splfy the behavour of a olear te seres s really sutable to descrbe the observed data. Gve that addtvty represets a strog assupto ad that there are few ethods to test addtvty the authors propose a procedure for testg addtvty olear te seres aalyss. It s based o Geeralzed Lelhood Rato Volterra expaso ad oparaetrc codtoal bootstrap whle ts perforace ters of eprcal sze ad power ad coparsos wth other addtvty tests are ade wth help of the Mote Carlo sulatos. The proble of defg proved varace estators for the ordary rato estator uder the Mdzuo-Se saplg schee s dscussed by Jga Patel ad P. A. Patel the O No-Negatve ad Iproved Varace Estato for the Rato Estator uder the Mdzuo-Se Saplg Schee. Startg fro observato that accordg to varous studes t s hard to sgle out the best ad o-egatve varace estator fte populato they propose a Mote Carlo coparso ad suggest estator that perfors well tag o-egatve values (wth probablty ). The paper Estato Of Lfe-Tables uder Rght-Cesorg by Agesza Rossa deals wth a class of o-paraetrc estators of codtoal probabltes of falure pror xy gve survval to x uder the rado ad observable rght-

4 5 Edtor s ote cesorshp odel. For the proposed estators based o a specfc sequetal saplg schee soe applcato lfe-table aalyss s preseted. I the paper Full Iforato Effcet Estator of Fte Populato Varace devoted to estatg quadratc or hgher order fte populato fuctos Mao Kuar Srvastava Nata Srvastava ad Housla P. Sgh suggest a effcet desg based full-forato estator of fte populato varace. Together wth provdg the exact expresso of the estator varace ad ts relatve effcecy they also show that the proposed estator s accordg to the eployed crtera of ts perforace a eprcal cotext superor to ts copettors. Aother aspect of estato s dscussed by Dwaar Shula Naredra Sgh Thaur Sharad Patha ad Dlp Sgh Raput Estato of Mea Uder Iputato of Mssg Data Usg Factor-Type Estator Two-Phase Saplg. As the proble of o-respose s oe of the ost portat saple surveys several putato ethods ted to copesate for the ssg observatos usg the avalable oes. Ths paper presets soe way to deal wth the proble of ut o-respose the cotext of two-phase saplg. Two dfferet strateges of such saplg sub-saple ad depedet saple are copared uder puted data setup usg Factor-Type (F-T) estators. The results of sulato perfored over ultple saples show that the frst putato strategy s foud better tha the secod (but the secod desg s better tha frst). I A Noparaetrc Cofdece Iterval for At-Rs-of-Poverty-Rate Wocech Zelńs refers to hs earler paper whch he proposed a dstrbutofree cofdece terval for the at-rs-of-poverty rate (ARPR) that was defed as the percetage of populato wth coe saller tha 60% of populato eda of the adult-equvalet dsposable coe. A exaple of applcato of the costructed cofdece terval s gve ths paper. A set of fve artcles the secod part of ths ssue represet a array of topcs. I Motorg Worers Rettaces ad Beefts Ugada: The Statstcal Issues E.S.K. Muwaga-Zae presets the efforts that are uderway by the Cetral Ba ad Cetral Statstcs Offce Ugada to prove the regulatory ad otorg evroet the coutry order to provde credble forato o rettaces that are creasgly growg ters of scope ad portace. The approach used cludes the eactet of a ew law ad regulatos provg adstratve reportg ad carryg out surveys the aor rettg coutres ad Ugada soe ssues of collectg accurate ad tely data are dscussed. The proble of poverty ad coe dyacs s aalyzed by Natala Nehrebeca Teporal Aspects of Poverty Polad Betwee by Hazard Models usg pael data fro CHER (Cosortu of Household Paels for Europea Soco-Ecooc Research). A tedecy to persstet poverty alog wth geerally low household coe dyacs that perod was show based

5 STATISTICS IN TRANSITION-ew seres Deceber o the rate of ext fro ad etry to poverty whle accoutg for both observed ad uobserved heterogeety of dvduals. I Kurtoss of a Rado Vector Specal Types of Dstbutos Katarzya Budy ad Ja Tatar attept to geeralze defto of urtoss for the ultdesoal case ad prove ts essetal propertes. The geeralzed characterstc appled the sgle-deso case has the sae propertes as urtoss that s ow the lterature o sgle-desoal rado varables. The bass of coducted cosderatos s the defto of the power of a vector space wth the scalar product. A software relablty growth odel to study the cobed effect of creasg error detecto ad decreasg error troducto rate uder perfect debuggg s proposed by S. Chatteree L.N. Upadhyaya J.B. Sgh ad S. Nga Cobed Effect of Fault Detecto ad Fault Itroducto Rate o Software Relablty Modellg. The odel s developed based o o hoogeeous Posso process (NHPP). It ca be used to estate ad predct the relablty as well as the cost of a software product soe real lfe data has bee used to valdate the proposed odel ad to show ts usefuless. Soe uavodable drawbacs data arrageets such as overlappg aog groups of observatos (e.g. characterzed by socal deographc or coe sources categores) that ay create proble wth decoposto of G ad re-rag dces to aalyse potetal redstrbuto effects ad the ufaress of a tax systes s dscussed by Achlle Verzz Applyg the Hadaard Product to Decopose G Cocetrato Redstrbuto ad Rerag Idexes. Eployg the so called atrx Hadaard product ad showg how wth group across ad betwee groups ad trasvarato copoets ca be wrtte atrx copact fors author also deostrates how the sgs of Atso-Plotc-Kawa re-rag dex copoets ca be aalysed ad splt. Włodzerz OKRASA Edtor--Chef

6 56 Edtor s ote ACKNOWLEDGEMENTS TO REFEREES FOR 009 The Edtoral Board wshes to tha the followg referees who have gve ther te ad slls to the Statstcs Trasto ew seres durg the perod of the year 009 Alesadra Baszczyńsa Uversty of Łódź Polad Jace Bałe Uversty of Łódź Polad Katarzya Boloe- Lasoń Uversty of Łódź Polad Des Coffe Uversty of Irelad - Mayooth Irelad Czesław Doańs Uversty of Łódź Polad Ala Jędrzecza Uversty of Łódź Polad Ce Kedlar Hacettepe Uversty of Aara Turey Jerzy Korzeows Uversty of Łódź Polad Jerzy T.Kowales Uversty of Łódź Polad Ncholas T.Logford Popeu Fabra Uversty Spa George Meexes Arstotle Uversty of Thessalo Greece Adrze Młoda Statstcal Offce Pozań Aad D.Al-Nasser arou Uversty of Irbd Jorda Adrze Ochoc Uversty of Cardal Stefa Wyszyńs Warsaw Polad Włodzerz Orasa Cetral Statstcal Offce of Polad ad Uversty of Cardal Stefa Wyszys Warsaw Polad Walety Ostasewcz Wrocław Uversty of Ecoocs Polad Ias Papadtrou Arstotle Uversty of Thessalo Greece Dorota Peasewcz Uversty of Łódź Polad Waldear Popńs Cetral Statstcal Offce Polad Agesza Rossa Uversty of Łódź Polad Dvaar Shula Dr.H.S.Gaur Uversty of Sagar Ida Meeash Srvastava Dr.B.R. Abedar Uversty (forerly Agra Uversty)Ida Grażya Trzpot Acadey of Ecoocs Katowce Polad Jausz Żądło Acadey of Ecoocs Katowce Polad

7 STATISTICS IN TRANSITION-ew seres Deceber STATISTICS IN TRANSITION-ew seres Deceber 009 Vol. 0 No. pp. 57 SUBMISSION INFORMATION FOR AUTHORS Statstcs Trasto ew seres (ST) s a teratoal oural publshed otly by the Polsh Statstcal Assocato (PTS) ad the Cetral Statstcal Offce of Polad o a quarterly bass (durg t was ssued twce ad sce 006 three tes a year). Also t has exteded ts scope of terest beyod ts orgally prary focus o statstcal ssues pertet to trasto fro cetrally plaed to a aret-oreted ecooy through ebracg questos related to systec trasforatos of ad wth the atoal statstcal systes world-wde. The ST-s sees cotrbutors that address the full rage of probles volved data producto data dsseato ad utlzato provdg teratoal couty of statstcas ad users cludg researchers teachers polcy aers ad the geeral publc wth a platfor for exchage of deas ad for sharg best practces all areas of the developet of statstcs. Accordgly artcles dealg wth ay topcs of statstcs ad ts advaceet as ether a scetfc doa (ew research ad data aalyss ethods) or as a doa of foratoal frastructure of the ecooy socety ad the state are approprate for Statstcs Trasto ew seres. Deostrato of the role played by statstcal research ad data ecooc growth ad socal progress (both locally ad globally) cludg better-fored decsos ad greater partcpato of ctzes are of partcular terest. Each paper subtted by prospectve authors are peer revewed by teratoally recogzed experts who are guded ther decsos about the publcato by crtera of orgalty ad overall qualty cludg ts cotet ad for ad of potetal terest to readers (esp. professoals). Mauscrpt should be subtted electrocally to the Edtor: [email protected]. followed by a hard copy addressed to Prof. Wlodzerz Orasa GUS / Cetral Statstcal Offce Al. Nepodległośc 08 R Warsaw Polad It s assued that the subtted auscrpt has ot bee publshed prevously ad that t s ot uder revew elsewhere. It should clude a abstract (of ot ore tha 600 characters cludg spaces). Iqures cocerg the subtted auscrpt ts curret status etc. should be drected to the Edtor by eal address above or [email protected]. For other aspects of edtoral polces ad procedures see the ST Gudeles o ts Web ste:

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9 STATISTICS IN TRANSITION-ew seres Deceber STATISTICS IN TRANSITION-ew seres Deceber 009 Vol. 0 No. pp NONPARAMETRIC BOOTSTRAP TEST FOR AUTOREGRESSIVE ADDITIVE MODELS Luca Bagato Atoo Puzo ABSTRACT Addtve autoregressve odels are cooly used to descrbe ad splfy the behavour of a olear te seres. Whe the addtve structure s chose ad the odel estated t s portat to evaluate f t s really sutable to descrbe the observed data sce addtvty represets a strog assupto. Although lterature presets extesve developets o addtve autoregressve odels few are the ethods to test addtvty whch are geerally applcable. I ths paper a procedure for testg addtvty olear te seres aalyss s provded. The ethod s based o: Geeralzed Lelhood Rato Volterra expaso ad oparaetrc codtoal bootstrap (Jaqg ad Qwe 00). Ivestgato o perforace ( ters of eprcal sze ad power) ad coparsos wth other addtvty tests proposed by Che et al. (995) are ade recurrg to Mote Carlo sulatos. Key words: Addtve odels; Geeralzed Lelhood Rato; Volterra expaso; Bootstrap.. Itroducto Addtve odels face the trade-off betwee the sspecfcato proble ad terpretablty. It s ot surprsg f such tools are cooly used the statstcal applcatos to splfy data aalyss. Ths wde faly of odels ebodes a ey splfyg assupto that soe scale covarate effects are separable. Furtherore addtve structures allow us to overcoe the so-called proble of the curse of desoalty (Bella 96) by deso reducto. I detal addtve autoregressve odels appled to te seres aalyss assue that codtoal expectato fucto of the depedet varable t ca be wrtte as the su of sooth ters the lagged varables: Dparteto d Metod Quattatv per le Sceze Ecooche Azedal Uverstà degl Stud d Mlao-Bcocca e-al: [email protected]. Dparteto d Ipresa Culture e Socetà Uverstà d Cataa e-al: [email protected].

10 60 L. Bagato A. Puzo: Noparaetrc bootstrap [ ( ) ( x x )] ( x ) ( x ) E. () t t t p p The advatages of usg addtve odels for olear autoregresso are based o several reasos. Frst they are easer to terpret because they do ot volve teractos. Secodly ay crcustaces they ca provde adequate approxatos for ay applcatos. Thrdly uder the addtvty assupto uvarate soothg techques ca be used drectly oparaetrc estato resultg to a ore coprehesve estate. I addto uder addtvty the olear cotrbuto of each lagged varable to the respose varable ca be easly see; t ca be dsplayed graphcally ad soe cases ca be terpreted. As regard the proble of coputg the addtve copoets ay algorths le for exaple bacfttg (Bua et al. 989) ad argal tegrato (Lto ad Nelse 995) have bee provded ad proved. Whe the addtve structure s chose ad the odel estated t s portat to evaluate f t s really sutable to descrbe the observed data sce however addtvty represets a strog assupto. The questo arses fro the sspecfcato proble whch leads to wrog coclusos ad erroeous forecastg. The dagostc checg stage s ot erely to detere whether there s evdece of lac of ft but also to suggest ways whch the odel ay be odfed whe ths s ecessary. There are two basc ethods for odel valdato: overfttg ad dagostc checs appled to the resduals. Ths paper focuses o the overfttg approach where the odel s delberately overparaeterzed a way t s expected to be eeded ad a aer such that the etertaed odel s obtaed by settg certa paraeters the ore geeral odel at fxed values usually zero (Box ad Perce 970). Ths tradtoal approach aly based o paraetrc assuptos cossts of usg a large faly of paraetrc odels uder the alteratve hypothess. The plct assupto s that the large faly of paraetrc odels specfes the for of the true uderlyg dyacs correctly. However ths s ot always warrated ad leads aturally to a oparaetrc alteratve hypothess. Naturally the probles crease whe also the ull hypothess s oparaetrc (addtve structure ths specfc case). Although extesve developets o oparaetrc estato techques there are few geerally applcable ethods for testg addtvty (see Che et al. 995). Our proposed procedure that coes o top of the procedures proposed by Che et al. (995) s based o the Geeralzed Lelhood Rato (GLR) whch s a geerally applcable tool for testg paraetrc hypotheses agast oparaetrc alteratves. A exteso for usg such a procedure to the oparaetrc (addtve) ull hypothess case wll be ade. Although the GLR ethod has bee developed for depedet data the dea ca be appled to te seres data. I fact t s expected that uder xg codtos the results should also hold for the depedet data (Jaqg ad Qwe 00). The paper s orgazed as follows: a troducto to the GLR ethod ad oparaetrc codtoal bootstrap s preseted Secto ; the proposed p p

11 STATISTICS IN TRANSITION-ew seres Deceber procedure for testg addtvty s descrbed Secto ad the last secto t s appled to addtve ad oaddtve odels ofte used the te seres lterature.. The geeralzed lelhood rato Before troducg GLR t s worth to reeber the classc axu lelhood rato test whch s geerally applcable to ost paraetrc hypothesstestg procedure. The fudaetal property that cotrbutes to the success of the axu lelhood rato tests s that ther asyptotc dstrbutos uder the ull hypothess are depedet of usace paraeters. Ths property was referred to as the Wls pheoeo by Fa et al. (00). Assug such a property oe ca detere the ull dstrbuto of the lelhood rato statstc by usg ether the asyptotc dstrbuto or the Mote Carlo sulato by settg usace paraeters at soe ftted values. The latter s also referred to as the paraetrc bootstrap. The questo arses aturally whether the axu lelhood rato test s stll applcable to the probles wth oparaetrc odels as alteratve. Frst oparaetrc axu lelhood estators (MLE) usually do ot exst. Eve whe they exst they are hard to copute. To tgate these dffcultes the axu lelhood estator uder the alteratve hypothess ca be replaced by ay reasoable oparaetrc estator. Ths s the essece of the geeralzed lelhood rato. Let f be the vector of fuctos of a terest ad η be the vector of usace paraeters. Suppose that the logarth of the lelhood of a gve set of data s ( fη). Gve η a good oparaetrc estator fˆ η ca be obtaed. The usace paraeters η ca be estated by the profle lelhood by axzg ( f ˆ η η) wth respect to η resultg the profle lelhood estator ηˆ. Ths gves the profle lelhood ( fˆˆ η ηˆ ) whch s ot the axu lelhood sce fˆ η ˆ s ot a MLE. Now suppose that we are terested testg whether a paraetrc faly f θ fts a gve set of data. Forally the ull hypothess s H : f θ () f 0 θ Θ ad we use the oparaetrc odel f as alteratve. Let ˆ θ 0 ad ˆη 0 be the axu lelhood estators uder the ull odel () obtaed by axzg f η ηˆ s the axu lelhood uder the ull the fucto ( ) θ. The ( f ˆ θ 0 ) 0

12 6 L. Bagato A. Puzo: Noparaetrc bootstrap hypothess. The GLR statstc sply copares the log-lelhood uder the two copetg classes of odels: T ˆ ηˆ f ηˆ () ( ˆ ) ( ˆ ). f η Exaple. (Uvarate oparaetrc odel) Let {( )} saple fro the oparaetrc odel: where { } ( ) θ 0 0 be a ε () ε are a sequece of..d. rado varables fro N ( 0 σ ) testg the sple lear regresso odel: H ( ) β β x H : ( x) β β 0 : x 0 0 x. Cosder (5) wth oparaetrc alteratve odel (). The the codtoal log-lelhood fucto gve s ( σ ) l( πσ ) ( ) (6) σ [ ]. I ths specfc case results f ησ ad θ ( β β 0 ). Ultately θ detfes a partcular odel cotaed the lear odel class. For a gve σ let ˆ () be for exaple the local lear estator based o the data {( )} whch s depedet of σ. Substtutg t to (6) the followg profle lelhood s obtaed: where RSS [ ( )] ( ˆ σ ) l( πσ ) RSS (7) σ ˆ - that ˆ σ RSS. Hece the profle lelhood s. Maxzg (7) wth respect to σ t results π RSS ( ˆ ˆ σ ) l. (8) Uder H 0 the axu lelhood estator ˆ ( ˆ β ˆ β ) The the profle lelhood uder the ull hypothess results π RSS 0 ( ˆ σ 0 ) l θ ca be obtaed. (9) ˆ0 θ 0 0

13 STATISTICS IN TRANSITION-ew seres Deceber where RSS [ ( ˆ ˆ )] β β 0 0 statstc () ca be obtaed as follows:. Usg quattes RSS 0 ad RSS the RSS0 T l. RSS (0) Ths s a GLR test statstc. As wth paraetrc ferece the GLR test does ot have to use the true lelhood. For exaple the test statstc T Exaple. apples to proble (5) whether ε s orally dstrbuted or ot. The oralty assupto s sply used to otvate the procedure. Slarly the GLR statstc does ot have to requre the MLE uder the ull hypothess. Such cosderatos suggested usg the GLR ethod for testg addtvty. I fact ether MLE or error dstrbuto assuptos are ade. I the ext secto t wll be show how to provde the alteratve structure to copare wth the addtve oe. I order to do ths the Volterra expaso wll be used. Furtherore to utlze the GLR statstc the dstrbuto uder the ull hypothess eeds to be provded. The questo arses aturally whether the asyptotc ull dstrbuto depeds o the usace paraeter uder the ull hypothess aely whether the Wls pheoeo cotues to hold for the GLR test. For a uber of odels ad a uber of hypotheses studed by Fa et al. (00) t has bee show that the Wls type of results cotue to hold. Such authors are ot able to show that Wls type of results hold for all probles but ther results dcate that such a pheoeo holds wth suffcet geeralty.. A ew test for autoregressve addtvty wth The Addtve AutoRegressve (AAR) odel s defed as follows: ( ) ε t t t p t () ( ) c ( ) ( ) t t p t p t p () where c s a costat p are uvarate uow fuctos ad the ε t are depedet ad detcally dstrbuted (..d.) wth ea 0 ad varace σ. Furtherore ε s assued depedet of t { t } for ay t. To esure detfablty of the addtve copoet fuctos t s assued E [ ( t )] 0 for all p. The tercept c E( t ) s typcally estated by

14 6 L. Bagato A. Puzo: Noparaetrc bootstrap /. Techcally the..d. assupto of the errors ay be weaeed whe other theoretcal exploratos are ade. However as well ow a whte ose process s o loger a pertet buldg bloc for olear odels as t s portat to loo for easures beyod the secod oets to characterze the olear depedece structure. Oce estated the () uder the addtvty assupto () the obvous questo s whether such a odel s approprate to descrbe the uderlyg structure. I order to deal wth ths odel valdato the addtve (ull) hypothess p ( ) c ( ) H 0 : t t p t () wll be copared wth the (alteratve) hypothess that the codtoal ea has oe ore geeral autoregressve structure say H : ( t t p ). () The coparso wll be ade through the GLR statstc whch s utlzed here the ost geeral case that s whe also the ull hypothess s oparaetrc. Obvously the odel valdato procedure (overfttg techque) eeds to estate a very geeral odel uder the alteratve hypothess. The a proble relates to the defto of a odel ore geeral tha the addtve oe but ot affected by the curse of desoalty. The Volterra expaso allows us to overcoe such a dffculty (Che et al. 995). I partcular through the Volterra expaso a autoregressve odel ca be rewrtte the followg way t where p u c ( ) t φ u tu p u u t p p u v ε t φ uv tu tv p p ( t u ) φuvt ut v φuvwt ut vt w ε t u u v p φ uvw u v w tu tv tw u v w ε ( ) φ φ φ. t u u t u uu t u uuu t u t (5) Obvously the case u v w s excluded fro the suato of the thrdorder ter (5). It s clear fro expresso (5) that f the odel s addtve the all the coeffcets of the hgher-order ters the equato should be zero. Furtherore defg uv ( ) φ φ ( ) φ ( ) t u t v uv t u t v uuvv t u t v uuuvvv t u t v

15 STATISTICS IN TRANSITION-ew seres Deceber the expresso (5) ca be rewrtte as follows: p p p t c u ( t u ) uv( t ut v ) uvw( t ut vt w ) εt. (6) u u v u v w Such a result suggests us a sple approxate for to apply for the alteratve hypothess (). I partcular f for exaple the (6) s trucated to the secod suato the hypothess () ca be assued as: H : p p ( t t p ) c u ( t u ) uv( t ut ). (7) v u u v Ths forulato allows to estate a odel that cotas p p uvarate addtve fuctos. Although such a test s lted to the frst-order cross-product ters t should have acceptable power agast a large class of oaddtve odels. I the ext Subsecto the ethod for fdg the dstrbuto of the GLR statstc wll be provded... The codtoal bootstrap test The proposed step-procedure for testg addtvty that wll be called fro ow o as Codtoal Bootstrap Test s descrbed what follows. I. The two odels respectvely uder the ull ad the alteratve hypotheses () ad () are estated ad the GLR statstc T s calculated. II. The oparaetrc codtoal bootstrap (Jaqg ad Qwe 00) s appled:. Geerate the bootstrap resduals { ε * t } of the eprcal dstrbuto of the cetred resduals { ˆ ε t ˆ ε } fro the alteratve odel where εˆ s the average of { εˆ t }. A assupto about the dstrbuto error s ade; thus for exaple a erel desty estato ca be * * appled. Costruct the bootstrap saple: ad * ( ) ˆ ( ) ε * ˆ ˆ t c t p t p t t t t p for tp.. Estate the addtve ad the alteratve odel based o the bootstrap saple: * * * {( )}. t t p t t p t p

16 66 L. Bagato A. Puzo: Noparaetrc bootstrap Calculate the GLR statstc p RSS T l RSS * p RSS0 RSS * RSS * 0 *. Repeat the above two steps B tes ad use the eprcal dstrbuto of { T * } as a approxato to the dstrbuto of the GLR statstc T uder H 0. T * greater tha the statstc T provded at pot I. III. The estated p-value of the test s the percetage of { }. Sulato study I order to evaluate the perforace of the proposed test ters of ts eprcal sze ad power a sulato study s perfored. Obtaed results are also copared wth sze ad power of three dfferet addtvty tests proposed by Che et al. (995): the codtoal ea test the Lagrage ultpler test ad the perutato test. The frst uses the local codtoal ea estator of Truog (99) ad eploys a procedure slar to the aalyss of varace. The secod apples the alteratg codtoal expectato (ACE) algorth of Brea ad Freda (985) to ft a addtve odel to the data; addtvty s the tested by eas of a Lagrage ultpler type test. The thrd procedure uses the ACE algorth as well but t fts peruted resduals to soe cross-product ters of the explaatory varables order to obta a referece dstrbuto for the test statstc. I order to ae the above-etoed coparso easer two subsets of odels cosdered Che et al. (995) are used here oe for sze cosderatos ad the other for power aalyss. The frst set cossts of the followg two addtve odels: t 0.8 t 0. t εt (8).5 t s( t ) εt. (9) t 0 These odels are used to study the behavour of the codtoal bootstrap test uder the ull hypothess of addtvty. They represet te seres odels cooly used uvarate aalyss. The lear AR () odel (8) s chose to esure that the proposed test wors well for ths sple case whle the slghtly ore coplcated odel (9) cotag a trgooetrc se fucto at lag s ofte used the te seres lterature to descrbe perodc seres (Lews ad Ray 99). The secod set cossts of the followg two oaddtve odels: *.

17 STATISTICS IN TRANSITION-ew seres Deceber ( 0. ) exp( 0. ) ε exp t t t t t (0) t t ( ) ε. t s t t () These odels are used to study the power of the codtoal bootstrap test. I partcular odel () s a fuctoal-coeffcet AR() wth a se fucto of lag (Che ad Tsay 99). Le Che et al. (995) for each of the odels (8)-() we have appled the proposed test to 00 realzatos each wth 00 observatos. The saple sze of 00 or larger s coo olear te seres aalyss especally whe usg oparaetrc ethods; deed t s ofte dffcult to obta a relable estate of the hgh-desoal surface whe the saple sze s sall. Accordg to Che et al. (995) the ovatos ε t are depedet N(0). I applyg the codtoal bootstrap test order to ae faster the procedure we use a value B00 ad cross-product ters of degree oe the Volterra expaso. For detals o the sulato factors used for the other three tests see Che et al. (995). Table shows the (sulated) eprcal dstrbuto fucto of the p-values for odels (8) ad (9) ad for each of the four cosdered tests. Table. Percetles of p-values of the oparaetrc bootstrap test uder the ull hypothess coparso wth the tests proposed by Che et al. (995). Probablty Codtoal bootstrap test Codtoal ea test Lagrage ultpler test Perutato test Model (8) Model (9) Model (8) Model (9) Model (8) Model (9) Model (8) Model (9) The graphcal couterpart of Table s also gve Fgure ; here odels (8) ad (9) are separately cosdered Fgure (a) ad Fgure (b) respectvely.

18 68 L. Bagato A. Puzo: Noparaetrc bootstrap Fgure. Percetles of p-values of the oparaetrc bootstrap test uder the ull hypothess coparso wth the tests proposed by Che et al. (995). Fgure s useful because t splfes the coparatve aalyss of perforace; deed as expected t s easy to see that the eprcal dstrbuto fucto of the p-values for both odels (8) ad (9) s always close to the ufor dstrbuto the ut terval [0] whch eas that the oal sze equals the eprcal oe. Ths slarty s strgly clear for the codtoal bootstrap test. I these ters the codtoal bootstrap test appears to be oe of the best whle the perutato test the worst. Table shows the percetages of reecto by the several tests uder dfferet sgfcace levels for odels (0) ad (). Table. Percetages of reecto by the codtoal bootstrap test uder the alteratve hypothess coparso wth the tests proposed by Che et al. (995). Probablty Codtoal bootstrap test Codtoal ea test Lagrage ultpler test Perutato test Model (0) Model () Model (0) Model () Model (0) Model () Model (0) Model () It ca be easly oted how all the tests have a good power agast the fuctoal-coeffcet autoregressve odel (). Ufortuately above all the perutato test ad the codtoal bootstrap test do ot have a good power agast the expoetal odel (0). Ths poor perforace of the two tests s

19 STATISTICS IN TRANSITION-ew seres Deceber uderstadable because the oaddtvty of the alteratve odel s hgherorder ters ad oly the sple cross-product ter t t has bee used the tests. I practce t ay be helpful to eploy several cross-product ters usg these tests. 5. Cocludg rears I ths paper a ew procedure for testg addtvty whch we defed codtoal bootstrap test has bee proposed by eas of Geeralzed Lelhood Rato Volterra expaso ad oparaetrc codtoal bootstrap. Ths procedure does ot requre ay strog assupto o ovatos. A sulated aalyss of perforace ters of eprcal sze ad power suggests that the codtoal bootstrap test s geerally relable uder the ull hypothess eve f t ay result low power whe the true alteratve odel s oaddtve hgherorder ters ad the cross-product ters cosdered the Volterra expaso are sall uber. Ths drawbac also shared by the perutato test proposed Che et al. (995) could be however overcoe by eployg several crossproduct ters the above sad expaso (aturally to the detret of the coputg te). The paper hts at soe further ssues; for exaple a sort of rule of thub order to select the sutable uber of cross-product ters the Volterra expaso could be terestg. I detal future wors wll be drected to pleetg forato crtero techques to select the correct odel sde a wde faly of odels resultg fro the Volterra expaso. REFERENCES BELLMAN R. 96. Adaptve Cotrol Process. Prceto: Prceto Uversty Press. BO G. E. P. ad PIERCE D. A Dstrbuto of resdual autocorrelatos autoregressve-tegrated ovg average te seres odels. Joural of the Aerca Statstcal Assocato 65() pp BREIMAN L. ad FRIEDMAN J. H Estatg optal trasforatos for ultple regresso ad correlato. Joural of the Aerca Statstcal Assocato 80(9) pp BUJA A. HASTIE T. ad TIBSHIRANI R Lear soothers ad addtve odels. The Aals of Statstcs 7() pp CHEN R. ad TSA R. S. 99. Fuctoal-coeffcet autoregressve odels. Joural of the Aerca Statstcal Assocato 88() pp

20 70 L. Bagato A. Puzo: Noparaetrc bootstrap CHEN R. LIU J. S. ad TSA R. S Addtvty tests for olear autoregresso. Boetra 8() pp FAN J. ZHANG C. ad ZHANG J. 00. Geeralzed lelhood rato statstcs ad Wls pheoeo. The Aals of Statstcs 9() pp HASTIE T. ad TIBSHIRANI R. J Geeralsed addtve odels. Lodo: Chapa ad Hall. JIANQING F. ad QIWEI. 00. Nolear te seres: oparaetrc ad paraetrc ethods. New or: Sprger. LEWIS P. ad RA B. 99. Nolear odellg of ultvarate ad categorcal te seres usg ultvarate adaptve regresso sples. I H. Tog ed. Deso Estato ad Models. Sgapore: World Scetfc pp LINTON O. ad NIELSEN J. P A erel ethod of estatg structured oparaetrc regresso based o argal tegrato. Boetra pp. 8() TRUONG. 99. A oparaetrc fraewor for te seres aalyss. New Drectos Te Seres Aalyss pp New or: Sprger.

21 STATISTICS IN TRANSITION-ew seres Deceber STATISTICS IN TRANSITION-ew seres Deceber 009 Vol. 0 No. pp ON NON-NEGATIVE AND IMPROVED VARIANCE ESTIMATION FOR THE RATIO ESTIMATOR UNDER THE MIDZUNO-SEN SAMPLING SCHEME Jga Patel ad P. A. Patel ABSTRACT Varous studes o varace estato showed that t s hard to sgle out a best ad o-egatve varace estator fte populato. Ths paper attepts to fd proved varace estators for the ordary rato estator uder the Mdzuo-Se saplg schee. A Mote Carlo coparso has bee carred out. The suggested estator has perfored well ad has tae o-egatve values wth probablty. Key words: Model-based estato Mote Carlo Sulato Rato estator Varace estato. Itroducto A aor proble a large coplex survey s the selecto of a varace estato procedure. Most of the basc theory developed the stadard saplg texts deals wth varace estato for lear estators ad therefore s ot applcable to coplex survey volvg rato ad coposte estato procedures. However these varace estators are ot free fro weaesses. Also these estators have ot corporated the auxlary forato. The regresso ad rato estators are wdely used survey practce. I the past ore atteto has bee gve to the rato estator because of ts coputatoal ease ad applcablty for geeral saplg desgs. May varace estators for the rato estator have bee proposed ad copared. The ost of the are desg-based see e.g. Rao (969) Rao ad Beegle (967) Rao (968) Rao ad Rao (97) Rao ad Kuz (97) Royall ad Eberhardt (975) Royall ad Cuberlad (978 98) Krews ad Charabarty (98) ad Wu (98) Departet of Statstcs Sardar Patel Uversty Vallabh Vdhyaagar-880 Guarat Ida e-al: [email protected]. Departet of Statstcs Sardar Patel Uversty Vallabh Vdhyaagar-880 Guarat Ida e-al: [email protected].

22 7 J. Patel P. A. Patel: O o-egatve aog others. The theoretcal coparsos of varous varace estators have bee ade by assug that the varables x ad y populato satsfy soe lear regresso odels. Several authors studed the estato of odelvarace of rato predctor uder varous odels see e.g. Muhopadhyay (996) ad refereces cted there. The ssue of varace estato for these wdely used estators has ot bee fally resolved. Studes by Wu (98) Wu & Deg (98) ad Deg & Wu (987) show that t s hard to sgle out a best varace estator; optalty depeds o the perforace crtero use. Ths artcle deals wth the estato of varace of the rato estator uder the Mdzuo-Se saplg schee whe auxlary forato s avalable. I secto we suggest varace estator for the rato estator. Secto presets the results of a Mote Carlo study that copares the suggested estator wth the stadard ad avalable estators. Fally our coclusos are gve Secto. Let U { N} be a fte populato ad let y ad x be the values of the study varable y ad a auxlary varable x for the th populato ut... N. If A U we wrte Σ A for Σ A ad ΣΣ A for ΣΣ A. We see y s to estate the varace of the rato estatorˆ R of the populato total p s x U y where p wth Σ U (... N ) o the bass of a saple s of fxed-sze draw accordg to a saplg desg p (s) wth postve cluso probabltes π P( s) ad π P( s) for every ad. E P ( ) ad V P ( ) deote the desg-expectato ad desg-varace of a estator. The varace of ˆ R suggested by Mdzuo (950) s gve by where for P ( ˆ ) Λ( s ) y Λ V ( s ) y... N R U U y Λ( s ) M s P s f M s P s f

23 STATISTICS IN TRANSITION-ew seres Deceber ad s s p P N M. Rao (97) proposed Λ Λ s s R y y s y s v π π ) ( ) ( ) ˆ ( as a ubased estator of ) ˆ ( R R V. Further he stated that a suffcet codto for ) ˆ ( R v to be postve s that 0 ) ( Λ s for all. Chaudhur (975) assues that the characterstc y ca tae egatve values whch case the above suffcet codto s ot vald. Therefore he suggested alteratve ubased estators by wrtg ( ) P R V ˆ to dfferet fors as < N N R P t t T N T t V ) )( ( ) ( ) ˆ ( ) ˆ ( Q V U R P ) ˆ ( R V U R P where ) Λ( s T ) Λ( s T y t ) ( t N T t t T t N T Q ad ) ( t N T t t T t N T R Hs suggested varace estators are the < s s R t t T N T t v π π ) )( ( ) ( ) ˆ (

24 7 J. Patel P. A. Patel: O o-egatve ) ˆ ( Q v s R π ) ˆ ( R v s R π Chaudhur (976 98) ad Chaudhur ad Arab (98) studed the proble of o-egatve varace estato ad proposed suffcet codtos for oegatvty for ther estators < s s s s x M x N x x C v ) ( ) ( < s s s s x M x M M x x C v v π < s s s s x M N x x x C v < s s s s x x M x x C v π where p y p y C ) ( ) / ( s p M s P N M ad s s x x However t should be oted that oe of the estators s of the for gve by Rao (979). Moreover these suffcet codtos caot be satsfed ether at all or except trval case whe sze easures are equal. Rao ad Vaya (977) suggested < 0 0 )) ( ( ) / ( ) / ( ) ( s p s P s P s p M x x C v v s ad Π < s s s x M x x C v

25 STATISTICS IN TRANSITION-ew seres Deceber whch satsfy Rao s (977) codto for the ecessary for of o-egatve quadratc ubased estators. Tracy ad Muhopadhyay (99) also addressed the sae proble aely o-egatve ubased varace estato for the Mdzuo strategy. Vaya et al. (995) exteded the results of Rao ad Vaya (977) for o-egatve ubased varace estato of quadratc fors ad partcular dscussed varous estators of varace of estators of the populato total.. The suggested estator Vallat et al. (000) has etoed that relatvely lttle research has bee drected toward dervg varace estators that explctly corporate both desg-based ad odel-based thg. I ths secto odel-desg-based estator s suggested. Assue that y y... yn are exchageable rado varables havg ot dstrbuto ξ (See Cassel et al. 977 p.0) ad that the relatoshp betwee y ad x s y Ε V C ξ ξ ξ βx ( ε / x ( ε / x ε ) 0 ) σ x ( ε ε / x x ) ρσ x x ( ) () where σ β > 0 ad ρ are the paraeters. Here N E ( ) V ( ) ad C ( ) deote ξ -expectato ξ -varace ad ξ -covarace ξ ξ ξ respectvely. The paraeter ρ the odel () was show to be a qute geerally redudat. (See e.g. Brewer ad Ta 990 ad Patel ad Shah 999). Heceforth we assue ρ 0.We try to ae as effcet use of auxlary forato as possble through odel. To fd a optal strategy (a cobato of saplg desg ad estator) drect zato of desgvarace V p (v) s possble but gve a odelξ we ca try to ze the atcpated varace (See Isa ad Fuller 98) AV ( v VR ) Eξ E p[( v VR ) ] [ Eξ E p ( v VR Clearly whe v s p-ubased the AV ( ) becoes AV ( v VR ) Eξ E p [( v V R ) ] )]

26 76 J. Patel P. A. Patel: O o-egatve Here the optalty s terpreted sese of zg E E [( v ) ] subect to E p ( v) VR RN. Uder odel () we have U E ( V ) ( σ β ) Λ( s ) x β Λ( s ) x x ξ R We ust fd v to ze E ξ E p v VR ) ] E p{ Vξ ( v)} U [( E { B ( v)} p ξ ξ p V R subect to Eξ E p ( v) Eξ ( VR ) where B ξ ( v) E ξ ( v VR ). Proceedg o the sae le as lea.5 of Cassel et al. (977) (See also Patel ad Shah 999) t s easy to show that y y y σ β ˆ ad β ˆ () x ( ) x x s are respectvely uque ubasedess ples s foud to be v OPT s pξ -ubased predctors of σ β ad β. As p- pξ -ubasedess the optal p-ubased predctor of V R s y Λ( s ) π 0 s y y Λ( s ) π 0 () where Λ( s ) x π 0 ad Λ( s ) x s π 0 ( ) Λ( s ) x x Λ( s ) x x s It s clear fro () that v OPT wll reflect the true paraeter value closely whe the best lear ft betwee y ad x goes through the org ad the resdual fro t are sall. Rear. The optal cluso probabltes are ot cosstet sce ΣΣ Uπ 0 ( ) π 0 Rear. It s ulely that a desg s chose solely for the purpose of optu estato of a quadratc fucto.

27 STATISTICS IN TRANSITION-ew seres Deceber Sulato The precedg varace estators are copared eprcally o atural populatos lsted Table of Appedx B. For coparso of the varace estators v v v v ad v OPT a saple of sze was draw usg Mdzuo-Se saplg schee fro each of the populatos ad these varace estators were coputed. These procedure s repeated M tes. For each varace estatev ts relatve percetage bas s calculated as the relatve effcecy as v V RB( v) 00 * V MSE( v ) RE ( v) MSE( v) M M where v v( ) MSE( v) ( v( ) V ) M M Table reports the values of RE of the estators v v v ad v OPT. The values of RB(%) ad probablty of tag egatve values are reported Table ad Table respectvely gve Appedx A. Rear. Aog v v ad v (suggested by Chaudhur see Secto ) v s better for ost cases. Moreover the perforaces of v ad v are slar. For these reasos the sulated results o these estators are ot preseted here. Rear. Aog the estators v 0 v 0 ad v v 0 s better tha v ad v s better tha v 0. For ost of the populatos ther RBs are egatve dcatg that these estators are uder estatg the true varace. Also these estators have tae frequetly the egatve values for ost of the populatos. Moreover v 0 s cosstetly poorer tha v both crtera. I short the estators v 0 v 0 ad v are shattergly bad ad therefore the correspodg results are otted fro the respectve tables.

28 78 J. Patel P. A. Patel: O o-egatve Table. RE uder Mdzuo Saplg Popl. v v v v v OPT Tables lead to the followg coets The estators v v ad v are coparable aog each other fro RE pot of vew. Aog these estators v has saller absolute RBs ad has tae egatve values for a very few populatos wth eglgble probabltes. Overall v s the best v the ddle ad v the worst for ost cases. v OPT has saller MSE but have bgger bas ( agtude). Eprcally v OPT s the oly o-egatve estator for all the populatos. The scatter plot of the populatos ad reveals that a lear odel y β x ε ght be approprate ad the relatoshp betwee y ad x s strog. The populatos have the varace structurev ( y ) x whereas the populatos 7 9 ad have the

29 STATISTICS IN TRANSITION-ew seres Deceber varace structurev ( ) x y. The relatoshp betwee y ad x s curvlear for the populatos 8 ad 0 whereas for the populatos ad 0 o systeatc patter s foud though the correlato betwee y ad x s oderate to hgh. Clearly the populatos ad 0 have fulflled the requreets for the v OPT estator gve (). Obvously for these populatos v OPT has perfored better tha the other (except oe case).. Coclusos Based o the eprcal study prevous secto we arrve at the followg coclusos. ) We ca ra the perforace of v v ad v as v v v where eas better tha wth respect to all crtera cosdered here. Thus the estators v (wth α β 0 ) suggested by Chaudhur (98) perfored very well for the populato havg oderate to hgh correlato betwee y ad x. ) It s clear that v OPT wll reflect the true varace clearly whe the best lear ft goes through the org ad the resdual fro t are sall. v OPT wll perfor badly f the true odel devates fro the assued odel.

30 80 J. Patel P. A. Patel: O o-egatve Appedx A Table. RB(%) uder Mdzuo Saplg. Popl. v v v v v OPT

31 STATISTICS IN TRANSITION-ew seres Deceber Table. Probablty of tag egatve values Popl. p p p p p OPT

32 8 J. Patel P. A. Patel: O o-egatve Appedx B Table. Study Populato Popl. N CV(x) CV(y) ρ(xy) Source x y D. Guarat p.7 Moey Supply GNP D. Guarat p. Labor put (per thousad persos) Real gross product llos of NT ($) Murthy(967) p.99 area uder wheat (96) area uder wheat (96) D. Guarat p.8 Moey Supply GNP D. Guarat p.7 Aerospace dustry sales Defese budget outlays Murthy(967) p.99 area uder wheat (96) area uder wheat (96) D. Guarat p.5 GNP Merchadse ports Murthy(967) p.8 Fxed Captal Output for Factores D. Guarat p.8 Real dsposable coe per capta ($) Per capta cosupto of chces (lbs) D. Guarat p.5 Wage coe Cosupto D. Guarat p.8 Coposte real prce of chce substtutes per lb weghted avg.of x to x D. Guarat p.8 Real retal prce of beef per lb Per capta cosupto of chces (lbs) Per capta cosupto of chces (lbs) Murthy(967) p.8 Fxed Captal Output for Factores a rego D. Guarat p.8 Real retal prce of por per lb Per capta cosupto of chces (lbs) D. Guarat p.0 Log-ter terest rate (%) Noal oey crores of rupees Murthy(967) p.8 uber of persos (95) o. of cultvators D. Guarat p.6 Real Captal put (llos of NT$) Real gross product llos of NT ($) Murthy(967) p.8 area sq.les o. of cultvators Murthy(967) p.99 cultvated area (96) area uder wheat (96) Murthy(967) p.8 uber of persos (96) o. of cultvators D. Guarat p.79 Icoe Savgs D. Guarat p.0 Expected or atcpated flato rate (%) at te t Actual rate of flato ( %) at te t

33 STATISTICS IN TRANSITION-ew seres Deceber Acowledget We would le to tha aoyous referees ad Prof. Paral Muhopadhyay for helpful coets ad suggestos. REFERENCES BREWER K.R.W. ad TAM S.M. (990). Is the assupto of ufor traclass correlato ever ustfed? Australa Joural of Statstcs (). CASSEL C. M. SÄRNDAL C. E. ad WRETMAN J. H. (977). Foudato of Iferece Survey Saplg Joh Wley New or. CHAUDHURI A. (975). O soe ferece probles wth fte populatos ad related topcs survey saplg theory. Ecooc Statstcs Papers. No Uversty of Sydey. CHAUDHURI A. (976). A o-egatvty crtero for a certa varace estator. Metra CHAUDHURI A. (98). No- egatve ubased varace estators. Curret topcs survey saplg Acadec Press Ic CHAUDHURI A. ad ARNAB R. (98). O o- egatve varace estato. Metra 8. DENG L.. ad WU C.F.J. (987). Estato of the regresso estator. Joural of Aerca Statstcal Assocato GUJARATI D. N. (995). Basc Ecooetrcs McGraw-Hll Ic. ISAKI C. T. ad FULLER (98). Survey Desg uder the regresso superpopulato odel Joural of the Aerca Statstcal Assocato KREWSKI D. ad CHAKRABART R. P. (98). O the stablty of the acfe varace estator rato estato. Joural of Statstcal Plag ad Iferece MIDZUNO H. (950). A outle of the theory of saplg systes. Aals Isttute of Statstcs ad Matheatcs Toyo MONTGOMER D. C.; PECK E. A. ad VINING G. G. (00). Itroducto to lear regresso aalyss. Thrd Edto Joh Wley & Sos Ic. MUKHOPADHA P. (996). Iferetal Probles Survey Saplg New Age Iteratoal New Delh.

34 8 J. Patel P. A. Patel: O o-egatve MURTH M. N. (967). Saplg Theory ad Methods. Calcutta: Statstcal Publshg Socety. PATEL P. A. ad SHAH D. N. (999). Model-based estato for a fte populato varace. Joural of the Ida Statstcal Assocato 7() 7 5. RAO J. N. K. (968). Soe sall saple results rato ad regresso Estato. Joural of the Ida Statstcal Assocato RAO J. N. K. (969). Rato ad regresso estators ew developets Survey Saplg (N.L.Joh ad H. Sth eds.) Wley New or. RAO J. N. K. (979). O dervg ea square errors ad ther o-egatve ubased estators fte populato saplg. Joural of the Ida Statstcal Assocato RAO J. N. K. ad BEEGLE L. D. (967). A Mote Carlo study of soe rato Estators. Sahya Seres B RAO J. N. K. ad KUZIK R. A. (97). Saplg errors rato estato. Sahya Seres C 6 (97) 58. RAO J.N.K. ad VIJAAN K. (977). O estatg the varace saplg wth probablty proportoal to aggregate sze. Joural of the Aerca Statstcal Assocato RAO P.S.R.S. ad RAO J.N.K. (97). Sall saple results for rato estators. Boetra RAO T. J. (97). O the varace of the rato estator for the Mdzuo-Se Saplg schee. Metra RAO T. J. (977). Estatg the varace of the rato estator for the Mdzuo- Se saplg schee. Metra 0 5. ROALL R.M. ad CUMBERLAND W.G. (978). Varace estato fte populato saplg. Joural of the Aerca Statstcal Assocato ROALL R.M. ad CUMBERLAND W.G. (98). A eprcal study of the rato estator ad estators of ts varace. Joural of the Aerca Statstcal Assocato ROALL R.M. ad EBERHADT K.R. (975). Varace estates for rato estator. Sahya Seres C 7 5. TRAC D. S. ad MUKHOPADHA P. (99). O o-egatve ubased varace estato for Mdzuo strategy Pa. J. Stat. 0 () VALLIANT R DORFAM A.H. ROALL R.M. (000). Fte populato saplg ad ferece Wley New or.

35 STATISTICS IN TRANSITION-ew seres Deceber VIJAAN K. MUKHOPADHA P. ad BHATTACHARA S. (995). O o-egatve ubased estato of quadratc fors fte populato Australa Joural of Statstcs 7 () WU C. F. J. (98). Estato of varace of the rato estator. Boetra WU C.F.J. ad DENG L.(98). Estato of varace of the rato estator: A eprcal study. I Scetfc Iferece Data Aalyss ad Robustess.

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37 STATISTICS IN TRANSITION-ew seres Deceber STATISTICS IN TRANSITION-ew seres Deceber 009 Vol. 0 No. pp ESTIMATION OF LIFE-TABLES UNDER RIGHT-CENSORING Agesza Rossa ABSTRACT The paper deals wth a class of o-paraetrc estators of codtoal probabltes of falure pror xy gve survval to x uder the rado ad observable rght-cesorshp odel. The proposed estators are based o a specfc sequetal saplg schee. Applcato of the estators lfe-table aalyss s preseted. Key words: Lfe-table aalyss; o-paraetrc estato; rght-cesored data; sequetal saplg.. Itroducto Let T be a o-egatve rado varable represetg a durato te betwee two well-defed evets.e. a tal ad a fal evet (the secod evet s usually called a falure). Let T be defed o a probablty space (Ω A P). The survval fucto of T s the defed as F ( x) P( T > x) x R wth F (x) for x 0. Cosder a probablty that the falure occurs a te terval (xxy] gve T>x. Usg the actuaral otato such a probablty s usually deoted by y q x. Thus we have y q x > It ca be also expressed ters of F as P ( T x y T x). () For correspodece: Isttute of Statstcs ad Deography Uversty of Łódź ul. Rewoluc Łódź Polad; e-al: [email protected].

38 88 A. Rossa: Estato of lfe-tables Deotg F ( x) F ( x y) F ( x y). () F ( x) F ( x) y q x F ( x y) () F ( x) y p x we receve fro () the obvous forula y q x p. Thus y q x s a probablty of falure pror xy gve survval to x ad y p x s a probablty of survvg beyod xy gve survval to x (for y 0 ). I ay applcatos especally deographc ad actuaral studes a table of uercal values of y q x ad y p x for a certa selected values of x ad y are cosdered ost cooly the tegers. Typcally a coplete table shows values of y qx ad y p x for all teger values of x 0 ω ad for y where ω stads for a axal possble lfete. I such cases probabltes qx ad p x deoted hereafter by q x px are orgazed a table of ubers (see Table ). y x Table. Tabular survval odel x 0 ω q x q q q 0 ω p x p p p 0 ω Fro probabltes y qx ad y p x varous lfe-table characterstcs are derved such as ubers of lves ad falures a cohort group expected lfetes tes of exposure etc.. Two cesorshp odels I ay real-lfe stuatos durato tes T T T for dvduals caot be fully observed. There ca be oe or ore rado evets (other tha the falure) occurrg of whch terates the observato of soe dvduals. I such cases t s sad that the observatos are rght-cesored. For stace very ofte t s possble to eep a study ope utl all the dvduals the saple

39 STATISTICS IN TRANSITION-ew seres Deceber have expereced the falure. Istead a perod of observato (follow-up) s chose ad dvduals are observed utl the ed of that perod. If a -th dvdual fals by the te of the aalyss the t yelds a true durato te T otherwse a cesorg te Z say s observed such that Z < T. If dvduals eter the study at rado tes the the Z 's are rado varables depedet of the T 's. Uder the rado cesorshp odel [see Efro (967)] t s usually assued that the T 's are d ad depedet of the Z 's. It s also assued that T s observed wheever T Z ad Z s observed wheever Z < T. I other words a par δ ) s observed where ( ( T Z ) () δ T Z ) (5) ( ad ( ) deotes a dcator fucto dcatg whether s rght-cesored ( δ 0 ) or ot ( δ ). However f cesorg echas s oly due to the terato of the observato perod the each Z represets the te elapsed fro eterg a - th subect to the study to the ed of the observato perod. I such cases the Z 's are fully observed ad oe observes ( Z ) for. We wll refer to ths specal type of the odel ()-(5) as the rado ad observable cesorshp odel. Throughout the rest of the paper we wll assue that the durato tes T as well as the cesorg tes Z are utually depedet cotuous rado varables the probablty space (Ω A P) ad that the T 's have a coo cuulatve dstrbuto fucto (cdf) F whereas the Z 's have a coo cdf G. These assuptos ply that varables defed () have a coo cdf expressed as where F F G G. H F G

40 90 A. Rossa: Estato of lfe-tables. Estato of codtoal probabltes uder the rado cesorshp odel Let us cosder a rado cesored saple of a fxed sze δ)( δ ) ( δ ). (6) ( Hereafter observatos ( δ ) wll be called falures whereas ( δ 0) wll be called losses. Let 0 x x x x y (7) 0 < < < J < J be a partto of the rage 0 y ] of observato to J tervals I ( x x ] for 0 J. 0 J [ 0 We wll cosder the followg statstcs defed for each. 0 D L ( I δ ) (8) ( I δ 0) (9) M ( > x ) (0) where D ad L represet ubers of falures ad losses a terval ( x x ] respectvely whereas M represets uber of tes at rs at the begg of the terval (.e. ubers of dvduals survvg beyod x ). Note that M 0 ples D 0 ad L 0. Table. Lfe-Table Statstcs Uder Rado Cesorshp Model I Itervals ( x x ] ( 0 x ] ( x x ] x ] ( J x J No. of survvors M M M M 0 J No. of falures D D D D 0 J No. of losses L L 0 L L J

41 STATISTICS IN TRANSITION-ew seres Deceber where The well-ow estators of the codtoal probabltes y q ad y x x y p x x are usually defed by eas of the statstcs (8)-(0). For stace the Stadard Lfe-Table Estators ecoutered the lterature [e.g. Berso ad Gage (950) Geha (965) Breslow ad Crowley (97) Daya (005)] are defed as y q x M D ˆ ad L y pˆ x qˆ () Both y qˆ x ad y pˆ x are ot defed f M 0. I such cases t s usually assued that y qˆ x y pˆ 0 x. It s also well-ow that the Stadard Lfe-Table Estators y qˆ x are egatvely based what dcates that y pˆ x are based postvely. I geeral both estators are asyptotcally based ad ot cosstet [see Breslow ad Crowley (97)]. Tag to accout all these dsadvatages etoed above we wll propose a ew class of estators of probabltes () ad () whch are ubased ad cosstet. The proposed estators are derved uder the odel of rado ad observable cesorshp ad are based o a specal type of sequetal saplg.. Estato of codtoal probabltes uder rado ad observable cesorg Cosder the sequece of observatos ( Z)( Z ). Let be a fxed teger ( ) ad y 0 be a fxed real value such that 0 < y 0 < sup{ y : H ( y) < } where H( ) deotes a coo cdf of the s. Suppose that dvduals arrve at rado to the study ad the observato perod terates f for dvduals oe observes > y0. Such a sequetal saplg leads to a saple y ( Z )( Z ) ( N Z ) N () where the saple sze N s a rado varable dstrbuted accordg to the egatve boal dstrbuto wth paraeters ad p H ( y 0 ). Let us cosder the partto (7) of the te terval [ 0 y 0 ] ad defe the followg statstcs x

42 9 A. Rossa: Estato of lfe-tables > > N J J x Z x R ) ( () > N J J x M 0. ) ( () I the specal case there s N M 0 ad.. M J Defto. Uder the odel of rado ad observable cesorshp the lfe-table estators of the codtoal probabltes x y q ad x y p ( J 0 ) tae the for x y R M R q ~ (5) ad x y x y q p ~ ~. (6) where x x y. Proposto. The estators (5) ad (6) are ubased ad cosstet estators of the codtoal probabltes x y q ad x y p. Ther varaces satsfy the followg equvalece ( ) ( ) ` ~ ~ x y x y x y x y R q R q p q E E V V (7) where the expectatos R E ad R E ca be expressed as q du u u q p R 0 ) ( E q du u u q q p R 0 ) ( E

43 STATISTICS IN TRANSITION-ew seres Deceber ad y p x p F ( x ) F ( x ) H ( y0 ) F ( x ) G ( x y q ) x q y p x p. The proof of the proposto was gve by Rossa (005) pp It follows fro (7) that varaces of ~ ad ~ ca be estated by eas of the followg expresso ˆ y q x y p x ( ~ ) ˆ ( ~ ) ~ ~ q V p q q. V (8) y x y x y x R y x R 5. A uercal exaple To llustrate detal the saplg schee ad the estators proposed a hypothetcal study wll be cosdered. Assue that the subect of observato s the te T elapsed fro the ssuace of a lfe-surace polcy up to the death of a -th sured perso ( years) ad let Z deote the te elapsed fro the ssuace of hs/her polcy up to the terato of the follow-up study. Due to rght-cesorg varables T T are possbly uobserved. However varables ( T Z ) as well as cesorg varables Z for are fully observed. We wll assue that persos arrve at rado to the study ad the follow-up perod terates whe for 5 sured persos we observe > y0. Suppose here that y 0 5 (years). Cosder a partto of the rage [ 0 y 0 ] to subtervals I ( x x ] where x 0 y 0. Table (colus ad ) cotas exeplary values of statstcs R x M x x 0 whch ca be observed uder such a saplg schee. The ext three colus preset estates of the codtoal probabltes () ad () derved fro the forulae (5) ad (6) for y as V ~ obtaed fro the forula (8). q x well as estates of ther varaces ( ) Table. Estates ~ ~ of probabltes x p x q 5 x p5 x q ad V ( ~ ) q 5 x

44 9 A. Rossa: Estato of lfe-tables x 0 R 5 x M 5 x q 5 x p x ~ ~ 5 V ˆ ( ~ ) q 5 x Dscusso I the paper two classes of o-paraetrc estators of codtoal probabltes y q x P ( T x y T > x) ad y p x P ( T > x y T > x) are proposed derved uder the so-called rado ad observable cesorshp odel. The proposed estators are ubased ad cosstet as opposed to the well-ow Stadard Lfe-Table Estators. Both classes of estators are based o a specal sequetal saplg schee. I ths schee a teger ad a postve value y 0 such that y 0 < sup{ y : H ( y) < } have to be fxed advace where H s a cdf of ( T Z). Note that usually t s ot dffcult to choose a proper value of y 0 eve f the fucto H s uow. It s suffcet to ow the axal possble values t ad z say of the durato ad cesorg tes respectvely. The for ay y0 ( 0( t z)) ad the codto y sup{ : ( ) 0 < y H y < } s satsfed.

45 STATISTICS IN TRANSITION-ew seres Deceber REFERENCES BERKSON J. & GAGE R Calculato of Survval Rates for Cacer. Proc. of the Staff Meetgs of the Mayo Clc 5 pp BRESLOW N. E. & CROWLE J. J. 97. Large Saple Study of the Lfe Table ad Product Lt Estates uder Rado Cesorshp. Aals of Statstcs pp DAA S Lfe Table (Survval) Aalyss to Geerate Pregacy Rates Asssted Reproducto: Are We Overestato Our Success Rates? Hua Reproducto 0 5. EFRON B The Two-Saple Proble wth Cesored Data. Proceedgs of Ffth Bereley Syposu o Matheatcal Statstcs ad Probablty pp Uversty of Calfora Press Bereley. GEHAN E. A A Geeralzed Wlcoxo Test for Coparg Arbtrarly Sgle-Cesored Saples. Boetra 5 pp. 0. ROSSA A Estato of Survval Dstrbutos uder Rght-Cesorg ad Applcatos Uversty of Łódź ( Polsh).

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47 STATISTICS IN TRANSITION-ew seres Deceber STATISTICS IN TRANSITION-ew seres Deceber 009 Vol. 0 No. pp. 97 ESTIMATION OF MEAN UNDER IMPUTATION OF MISSING DATA USING FACTOR-TPE ESTIMATOR IN TWO-PHASE SAMPLING Dwaar Shula Naredra Sgh Thaur Sharad Patha ad Dlp Sgh Raput ABSTRACT I saple surveys the proble of o-respose s oe of the ost frequet ad wdely appearg whose soluto s requred to obta usg relevat statstcal techques. The putato s oe such ethodology whch uses avalable data as a source for replaceet of ssg observatos. Two-phase saplg s useful whe populato paraeter of auxlary forato s uow. Ths paper presets the use of putato for dealg wth o-respodg uts the setup of two-phase saplg. Two dfferet two-phase saplg strateges (subsaple ad depedet saple) are copared uder puted data setup. Factor- Type (F-T) estators are used as tools of putato ad sulato study s perfored over ultple saples showg the coparatve stregth of oe over other. Frst putato strategy s foud better tha secod whereas secod saplg desg s better tha frst. Key words: Estato Mssg data Iputato Bas Mea squared error (MSE) Factor Type (F-T) estator Two-phase saplg Sple Rado Saplg Wthout Replaceet (SRSWOR) Coprosed Iputato (C. I.).. Itroducto Let {... N} Ω be a fte populato wth as a varable of a terest ad (... N ) N a auxlary varable. As usual N Dwaar Shula Naredra Sgh Thaur Sharad Patha Deptt. of Matheatcs ad Statstcs H.S. Gour Uversty of Sagar Sagar(M.P.) INDIA P e-als:[email protected] [email protected]. [email protected].. Dlp Sgh Raput Govt. College Rehl Sagar (M.P.) INDIA.

48 98 D. Shula N. S. Thaur S. Patha D. S. Raput: Estato of N N are populato eas s assued ow ad uder vestgato. Sgh ad Shula (987) proposed Factor Type (F-T) estator to obta the estate of populato ea uder setup of SRSWOR. Soe other cotrbutos o Factor-Type estator slar setup are due to Sgh ad Shula (99) ad Sgh et al. (99). Wth uow the two-phase saplg s used to obta the estate of populato ea ad Shula (00) suggested F-T estator uder ths case. But whe few of observatos are ssg the saple the F-T estator fals to estate. Ths paper udertaes the proble of Shula (00) wth suggested putato procedures for ssg observatos. Rub (976) addressed three ssg observato cocepts: ssg at rado (MAR) observed at rado (OAR) ad paraeter dstrbuto (PD). Heta ad Basu (996) explaed the cocept of ssg at rado (MAR) ad troduced the ssg copletely at rado (MCAR). The preset dscusso s o MCAR wherever the o-respose s quoted. Rao ad Stter (995) dscussed a ew learzato varace estator that aes ore coplete use of the saple data tha a stadard oe. They have show ts applcato to ass putato uder two-phase saplg ad deterstc putato for ssg data. Sgh ad Hor (000) suggested a Coprosed Iputato (C-I) procedure whch the estator of ea obtaed through C-I reas better tha obtaed fro rato ethod of putato ad ea ethod of putato. Ahed et al. (006) desged several geeralzed structure of putato procedures ad ther correspodg estators of the populato ea. Motvato s derved fro these ad fro Shula (00) to exted the cotet for the putato setup. Cosder a prelary large saple ' S of sze ' draw fro populato Ω ' by SRSWOR ad a secodary saple S of sze ( < ) draw ether of the followg aers: ' Case I: as a sub-saple fro saple S (deoted by desg F ) as fg. (a) Case II: depedet to saple S ' (deote by desg F ) as fg. (b) wthout replacg S '.

49 STATISTICS IN TRANSITION-ew seres Deceber Populato (N) Populato (N) ' ' S ' x y r R x r C R S ' ' S ' x S x y r R x r C R x Fg. (a) [Case I F ] Fg. (b) [Case II F ] Let saple S of uts cotas r respodg uts (r < ) forg a subspace C C R ad ( r) o-respodg wth sub-space R S R R. For every R the y s observed avalable. For puted values are coputed. The th value source of putato for ssg data whe x s ' { x : S} ad { ' ' ' : S } ' ' ( ) x respectvely. x ' ' C R the y values are ssg ad x of auxlary varate s used as a C R. Assue for S the data x x are ow wth ea x ( ) ad

50 00 D. Shula N. S. Thaur S. Patha D. S. Raput: Estato of. F-T Iputato Strateges Two proposed strateges d ad d for ssg data uder both cases are : d : ( ) d : ( ) y f R ' y ( ) d A C x fbx (.) C yr r y ( ) ' r f R r ( A fb) x Cx y f R ' y ( ) d A C x fbxr (.) C y r r y ( ) ' r f R r ( A fb) x C xr Uder (.) ad (.) pot estators of are : ( ) ( A C) y y d ' x fbx r ; (.) ( ) ( A C) y y ( ) ' A fb x C x ' x fbx r ; (.) d ( ) r ' A fb x C x r where A ( )( ) ; B ( )( ) ; ( )( )( ) ( 0 < < ) s a costat... Soe Specal Cases () At ; A 0; B 0; C 6 ( ) ' x y d y r x C ad (.5) ( ) ' x y d y r r x () At ; A 0; B ; C 0 (.6) x ( ) y y r ' x d (.7)

51 STATISTICS IN TRANSITION-ew seres Deceber x r ( ) y y r ' x () At ; A ; B ; C 0 ( y ) d (.8) ' ( ) x f x d y r (.9) ' f x ( y ) (v) At ; A 6; B 0; C 0 ' ( ) x f x d y r (.0) ' f x ( y d ) y r ( y d ) y r. Propertes of Iputato Strateges (.) (.) Let B(.) t ad M(.) t deote the bas ad ea squared error (M.S.E.) of estator uder saplg desg t I II (or F F ). Large saple approxatos are: ( ); x r ( ); x ( ) y r e e ' ' ad x ( e ) e Usg two-phase saplg followg Rao ad Stter (995) ad the ' echas of MCAR for gve r ad we wrte: () Uder desg F [Case I ]: ' E ( e ) E( e ) E( e ) E( e ) 0 ; E( e ) δc ; E( e ) ' E( e ) δ C ; E e δ C ; E e e δ ρ E ( e e ) ( ) x δ C ( ) C C ; δ ρ C C ; ( ' E e ) e δ ρc C E ; E( e e ) δ C ; ' ' ( e e ) δ C ; E( e e ) δ C ; ; () Uder desg F [Case II ]: ' E ( e ) E( e ) E( e ) E( e ) 0 ; E( e ) δ C ; E( e ) ' E( e ) δ C ; E e δ C ; E e e δ ρ ( ) 5 x δ C ( ) C C ; ;

52 0 D. Shula N. S. Thaur S. Patha D. S. Raput: Estato of E where ( e e ) δ ρ C C ; ( ' e e ) 0 5 Rear.: Let E ; E( e e ) δ C ; E ' ( e e ) 0 ' ( e e ) 0 5 δ ' ; δ ' ; δ ; ' r N δ ' ; δ 5 ' r N N E ; A C fb A fb C θ ; θ ; θ ; θ ; A fb C A fb C A fb C A fb C C P ( θ θ ) ( θ θ ); ( θ θ θ θ θ θ ) P ( θ θ ); V ρ. C Theore.: Estators ( ) d could be expressed : y ad ( ) y ters of e ; ad ' [ { ee} ] ' y d [ { ee} ] r s r ' s Whle gorg ters E[ e e ] E e ( e ) for s > s ' ' ' () y d e P e e θ e θ e ee ee ( θ θ ) ' ' ' () e P e e θ e θ e e e e e ( θ θ ) d (.) (.) [ ] ' e r r 0... ad... whch s frst order of approxato [see Cochra (005)]. Proof: () y d y r ' ( A C) x fbx ( ) ' A fb x C x ' ' ( e )( )( ) θ e θ e θ e θ ' ' ' ' [ e P{ e e θ e θ e e e e e ( θ θ ) e e }] e () y d y r ' ( A C) x fbx ( ) r ' A fb x Cx r ' ' ( e )( )( ) θ e θ e θ e θ ' ' ' ' [ e P{ e e θ e θ e e e e e ( θ θ ) e e }] e

53 STATISTICS IN TRANSITION-ew seres Deceber Theore.: Bases of ( ) t d y ad ( ) t d y uder t I II (or desg F ad F ) up to frst order of approxato are: () [ ] ( )( ) I d C C C P y B ρ θ δ δ (.) () [ ] ( ) [ ] II d C C C P y B ρ δ δ θ δ θ 5 5 (.) () [ ] ( )( ) I d C C C P y B ρ θ δ δ (.5) (v) [ ] ( ) [ ] II d C C C P y B ρ δ δ θ δ θ (.6) Proof: () [ ] [ ] I d I d y E y B ( ) { } [ ] ' ' ' ' e e e e e e e e e P e e E θ θ θ θ ( )( ) C C C P ρ θ δ δ () [ ] [ ] II d II d y E y B ( ) { } [ ] ' ' ' ' e e e e e e e e e P e e E θ θ θ θ ( ) [ ] C C C P ρ δ δ θ δ θ 5 5 () [ ] [ ] I d I d y E y B ( )( ) C C C P ρ θ δ δ (v) [ ] [ ] II d II d y E y B ( ) [ ] C C C P ρ δ δ θ δ θ Theore.: Mea squared errors of ( ) t d y ad ( ) t d y uder desg F ad F up to frst order of approxato are: () [ ] ( )( ) [ ] I d C C P C P C y M ρ δ δ δ (.7) () [ ] ( ) [ ] II d C C P C P C y M ρ δ δ δ δ 5 5 (.8) () [ ] ( )( ) [ ] I d C C P C P C y M ρ δ δ δ (.9) (v) [ ] ( ) [ ] II d C C P C P C y M ρ δ δ δ δ (.0) Proof: () [ ] [ ] I d I d y E y M

54 0 D. Shula N. S. Thaur S. Patha D. S. Raput: Estato of ' E[ e ( )] P e e [ δ C ( δ δ )( P C PρC C )] () M [ y ] [ ] d E y II d II ' E[ e ( )] P e e [ C ( δ δ ) P C Pδ ρc C ] δ 5 5 () M [ y ] [ ] d E y I d I ' E[ e ( )] P e e [ δ C ( δ δ )( P C PρC C )] (v) M [ y ] [ ] d E y II d II [ C ( δ δ ) P C Pδ ρc C ] δ Theore.: Mu ea squared errors of ( desg F ad F are : () M[ M ( y) ] [ ( ) ] d δ I δ δ ρ S () M[ M ( y )] [ ( ) ] d δ δ δ δ ρ S y d ) t ad ( d ) t y uder whe P V (.) II 5 5 whe δ V ( δ ) P (.) 5 / δ 5 () M ( y )] [ ( ) ] d δ I δ δ ρ S (v) M[ M ( y d )] [ δ ( δ δ ) δ ρ ] S Proof: d dp whe P V (.) II whe δ V ( δ ) P (.) / δ C () [ M ( y d ) ] 0 P I C M[ M ( y )] [ ( ) ] d δ I δ δ ρ S d () [ M ( y d ) ] 0 δ II 5V δ dp M[ M ( y )] [ ( ) ] d δ II δ δ 5 δ 5 ρ S d C () [ M ( ) ] 0 P ρ dp y d I ρ V ad usg ths (.7) ( δ ) P ad usg ths (.8) C 5 V

55 STATISTICS IN TRANSITION-ew seres Deceber M[ M ( y )] [ ( ) ] d δ I δ δ ρ S d (v) [ M ( y d ) ] 0 P δ II V δ dp M[ M ( y d )] [ δ ( δ δ ) δ ρ ] S II ( δ ) Lea.0 [By Shula (00)]: F-T estator two-phase saplg (wthout putato) s ( ) ( A C) y y d w ( ) ' A fb x C x ' x fbx (.5) Wth optu MSE codtos uder desg F [Case I]: P V (.6) uder desg F [Case I]: P V ( δ ) (.7) ad optu MSE expressos [ M ( y ) ] V 0 [ ρ ( δ )] w I [ M ( y ) ] V [ ρ ( δ ) ] opt opt d (.8) d w II 0 (.9) where δ ' N N ; V [( y ) ( x ) ] ; 0; 0 E. Approprate Choce of for Bas Reducto B y d P( θ ) 0 I C ρcc [ 5 f 6 f ] 0 () [ ] 0 P ( ) ( ) ( ) If 0 ad (.) [( 5 f ) ( f 6 f ) ] / [( 5 f ) ( f 6 f ) ] / (.)

56 06 D. Shula N. S. Thaur S. Patha D. S. Raput: Estato of If 0 C C ρc θ ( ) 0 C V VfB AV (.) ( ) ( ) [ ] ( ) [ ] ( ) [ ] V f V f V f V (.) () [ ] 0 II y d B ( ) [ ] C C C P ρ δ δ θ δ θ If 0 P we have soluto as per (.) ad f ( ) C C ρ δ δ θ δ θ ( ) ( ) ( ) ( ) ( ) V f V V f V V δ δ δ δ ( ) ( ) 0 5 V f V δ δ (.5) () [ ] 0 I y d B provdes slar soluto as (). (v) [ ] 0 II y d B ( ) [ ] 0 C C C P ρ δ δ θ δ θ f 0 P we have soluto as per (.) ad f ( ) 0 C C ρc δ δ θ δ θ ( ) [ ] ( ) C fb A V C fb A δ δ δ ( ) ( ) ( ) ( ) ( ) V f V V f V V δ δ δ δ ( ) ( ) 0 V f V δ δ (.6) 5. Coparso of the Estators () ( ) [ ] ( ) [ ] I d I d S N r y M y M ρ Δ ( ) ( ) I I better tha s d d y y f r N > > Δ 0 whch s always true. () ( ) [ ] ( ) [ ] II d II d y M y M Δ ( ) ( ) 5 5 S ρ δ δ δ δ δ δ

57 STATISTICS IN TRANSITION-ew seres Deceber ( y d ) II s better tha ( y d ) II f Δ > 0 ( r) [ N ( ) ] ' ' r r N ' r > 0 whch geerates two optos as (A) whe ( r) > 0 > r ad (B) ' ' [ N ( r r) N ' r] > 0 f ' N [. e. N ] ' the [ N ( r) N r] > 0 N (sce N > 0 always) ( N )( N r) > 0 ( N ) > 0 N > ad N r > 0 N > r The ultate s N > > r whch s always true. () Δ [ M ( y d ) ] [ M ( y d ) ] I II ( δ δ )( δ δ ) ( δ δ δ δ δ δ δ δ ) ( δ δ ) ( d ) II s better tha ( y d ) I y f Δ 0 ρ > r where > ' ( N ) ' ( N r) ρ S < ρ < or < ρ < 6. Eprcal Study The attached appedx A has a artfcal populato of sze N 00 [see Appedx A] cotag values of a varable ad auxlary varable. Paraeters of ths are gve table 6.. Table 6.. Populato Paraeters S S ρ C C C V ρ C ' ' Uder desg-i we draw a prelary rado saple S of sze 0 to copute x ' ad further draw a rado saple S of sze 50 such that

58 08 D. Shula N. S. Thaur S. Patha D. S. Raput: Estato of ' S S by SRSWOR. The V s a stable quatty over te ad assued to be ow [see Reddy (978)]. Table 6.. Desg Optu Codto for MSE Three optu values of o oe codto I P V P δ 5 V /( δ δ 5 ) II P V P δ V /( δ δ ) Sulato The bas ad optu.s.e. of proposed estators uder both desgs are ' coputed through repeated saples as per desg. Coputatos are table 7. where effcecy loss easureet due to putato s as ( ) [ M ( y s ) ] t LI t y s wth Opt [ M ( y ) ] t Opt[ M ( y d ) ] s the optu ea squared w error of estator y s s d d d ; t I II t w wthout putato. For desg I ad II the sulato procedure has followg steps : ' ' Step : Draw a rado saple S of sze 0 fro the populato of N 00 by SRSWOR. ' Step : Draw a rado sub-saple of sze 50 fro S for desg I ad depedet rado saple 50 fro ( N ' ) for desg II. Step : Drop dow 5 uts radoly fro each secod saple correspodg to both I ad II. Step : Ipute dropped uts of by proposed ethods ad avalable ethods ad copute the relevat statstc. Step 5: Repeat the above steps tes whch provdes ultple saple yˆ yˆ... y ad ˆ for estators ( y d ) t ( y d ) t y ˆ based estates ( s ) t ( s ) t ( s ) t ( s ) t ( y d ) w. Step 6: Bas of ( ) t s [ s ] t ŷ s B ( yˆ ) ( yˆ ) s t

59 STATISTICS IN TRANSITION-ew seres Deceber Step 7: M.S.E. of ( s ) t s t Step 8: The effcecy coparsos are M ( yd) I Desg effcecy M ( yd) II M ( yd ) I Desg effcecy E M ( yd ) II M ( yd) I Estator effcecy M ( yd ) I M ( yd) Estator effcecy E M ( y ) 50000[ ] ŷ s M ( yˆ ) ( yˆ ) E 00 ; 00 E 00 ; II d II Table 7.. Bas ad Mea Squared Error 00 s t Opt () Desg F Desg F ( y ) ( y ) ( y ) ( d d d y ) d Bas MSE Bas MSE Bas MSE Bas MSE

60 0 D. Shula N. S. Thaur S. Patha D. S. Raput: Estato of Table 7.. Estator wthout Iputato [Lea.0] Optu - values Optu MSE Case - I Case - II Table 7.. Loss due to Iputato LI t [ ] Opt () Desg F Desg F LI ( ) I y d LI I ( y d ) LI ( ) II y d LI ( ) II y d

61 STATISTICS IN TRANSITION-ew seres Deceber 009 Table 7.. Effcecy Coparsos E E E E Opt () E E E E % 7.5% 9.% 9.97% % 75.% 8.% 9.0% % 7.57% 9.% 9.09% % 7.6% 0.6%.80% % 57.8% 5.58% 0.56% % 5.85% 6.79% 0.0% % 7.5% 9.% 9.97% % 75.% 8.% 9.0% % 7.57% 9.% 9.09% % 5.% 6.77% 0.% % 6.6% 6.5% 0.89% % 58.0%.5% 0.9% 8. Alost Ubased Iputato Methods Usg equatos of secto.0 we get Fro () ' ; '. 5 ; '. 86 ; ' Fro () ' ; '. 5 ; '. 86 ; '. 6 ; ' ; ' Fro () slar to (). Fro (v) ' ; '. 5 ; '. 86 ; '. 9 ; ' ; ' Usg these -values we ca ae proposed F-T putato strateges alost ubased. The best aog the wll be that havg the lowest.s.e. By ths we have opto to choose alost ubased estator wth a cotrol over ea squared error.

62 D. Shula N. S. Thaur S. Patha D. S. Raput: Estato of 9. Dscusso ad Cocluso The proposed estators are foud useful for stuato whe soe observatos are ssg the saple. As per table 7. for y ad y uder desg F the effcet perforace of both s foud whe ad 6.7. O these specfc choces the loss of effcecy wth respect to wthout putato s very low. Slarly for y ad y uder desg F the effcet perforace observed at It sees eve by adoptg putato the suggested estators are loosg a lttle ters of relatve.s.e. to the wthout putato usual F-T estator. Whle utual coparsos are table 7. the desg F s uforly effcet as F at all the optu -values over both suggested F-T strateges. Wth F the estator y s ore effcet tha y whereas wth F t does d ot hold uforly for all -optals. The y d uder F foud better whe ad 6.7. Oe ca get alost ubased estators also o choces The ost sutable wll be that whch has the lowest.s.e. So these suggested strateges are alost ubased wth a reducg cotrol over.s.e. also. d d d d d

63 STATISTICS IN TRANSITION-ew seres Deceber 009 Appedx A Populato (N 00)

64 D. Shula N. S. Thaur S. Patha D. S. Raput: Estato of REFERENCES AHMED M. S. AL-TITI O. AL-RAWI Z. ad ABU-DAEH W. (006): Estato of a populato ea usg dfferet putato ethods Statstcs Trasto COCHRAN W. G. (005): Saplg Techques Joh Wley ad Sos Ffth Edto New or. HEITJAN D. F. ad BASU S. (996): Dstgushg Mssg at rado ad ssg copletely at rado The Aerca Statstca RAO J. N. K. ad SITTER R. R. (995): Varace estato uder two-phase saplg wth applcato to putato for ssg data Boetrca RUBIN D. B. (976): Iferece ad ssg data Boetrca SHUKLA D. (00): F-T estator uder two-phase saplg Metro SHUKLA D. SINGH V. K. ad SINGH G. N. (99): O the use of trasforato factor type estator Metro 9 ( ) SINGH S. ad HORN S. (000): Coprosed putato survey saplg Metra SINGH V. K. ad SHUKLA D. (987): Oe paraeter faly of factor type rato estator Metro SINGH V. K. ad SHUKLA D. (99): A effcet oe paraeter faly of factor - type estator saple survey Metro SINGH V. K. ad SINGH G. N. (99): Cha type estator wth two auxlary varables uder double saplg schee Metro REDD V. N. (978): A study o the use of pror owledge o certa populato paraeters estato Sahya C

65 STATISTICS IN TRANSITION-ew seres Deceber STATISTICS IN TRANSITION-ew seres Deceber 009 Vol. 0 No. pp. 5 5 FULL INFORMATION EFFICIENT ESTIMATOR OF FINITE POPULATION VARIANCE Mao Kuar Srvastava Nata Srvastava Housla P. Sgh ABSTRACT Secod order or quadratc ad fte populato paraetrc fuctos ay be expressed as total of varable-values o pars of uts a derved populato. Recetly Stter ad Wu (00) utlzed ths approach for estatg varace uder odel calbrato. I ths paper a effcet desg based full-forato estator of fte populato varace has bee suggested. The exact expresso of ts varace ad ts relatve effcecy has also bee derved. Fally the proposed estator has bee show to be superor to ts copettors a eprcal vestgato. Key words: desg based estato; varace estato; Rao-Blacwellzato survey saplg; estato of polyoal fte populato fucto.. Itroducto I fte populato theory the proble of estatg quadratc or hgher order fte populato fuctos s a extesvely explored area of research. Effcet estators for fte populato varaces covarace betwee two respose varables or varace of lear estators are hghly desrable. Effcet estato of these quadratc fuctos have bee prohbtve (Stter ad Wu 00) because of coplex expressos of varaces of these estators. Lu (97 a) suggested several desg based estators of fte populato varace followg whch Chaudhur (978) oted that ay of Lu's estators soetes tae egatve values. He the suggested alteratvely few o-egatve estators ad dscussed ther statstcal propertes. Later Lu ad Thopso (98) showed oexstece of the best ubased estator of populato varace ad asserted Assocate Professor Departet of Statstcs Isttute of Socal Sceces Dr. BRA Uversty Agra-800.U.P. Ida eal : [email protected].. Assocate Professor Departet of Statstcs St. Joh's College Agra-800 U.P. Ida. eal: [email protected]. Professor School of Studes Statstcs Vra Uversty Ua-5600 M.P. Ida eal: hpsu@ redffal.co.

66 6 M. K. Srvastava N. Srvastava H. P. Sgh: Full forato that the adssblty holds uder certa restrctos. Swa ad Mshra (99) suggested ore effcet estator of populato varace as copared to Lu's ad Chaudhary's estators ad tred t o varous atural ad artfcal populatos. Ths estator stll suffers fro a drawbac that t taes egatve values occasoally. Muhopadhyay (978) suggested odel based predctors of populato varace followg the Royall s predcto approach. Muhopadhyay ad Bhattacharya (989) suggested optal estators of populato varace uder regresso superpopulato odels. Stter ad Wu (00) for the frst te suggested very effcet odel calbrato estators whch were asyptotcally desg ubased uder lear ad o-lear odels. They suggested pseudoeprcal lelhood estators also whch are free fro tag egatve values. Bayesa estato of fte populato varace was frst tae up by Ercso (969) hs poeerg wor o Bayesa estato survey saplg although the ssue of estato of populato varace was brefly dscussed. Subsequet refereces are Lu (97 b) Chaudhur (978) Zacs ad Soloa (98) Ghosh ad Meede (98 98) Vardea ad Meede (98) Ghosh ad Lahr (987) Lahr ad Twar (99) Datta ad Twar (99) Datta ad Ghosh (99) aog others. Haurav (966) was frst who attepted splfcato of the proble of estato of populato varace by expressg t as a total of a fucto over a ew populato whose eleets are ordered pars of uts of the orgal populato. Usg ths approach Lu ad Thopso (98) showed oexstece of best ubased estators of populato varace ad gave soe adssblty results. More recetly Stter ad Wu (00) aed the populato of pars as "sythetc populato" ad uder the Haurav's approach they dscussed odel calbrato ad pseudoeprcal lelhood estators whch were odel asssted ad asyptotcally desg ubased uder a geeral saplg desg. We have suggested a effcet estator aed as full-forato estator of populato varace for equal ad uequal probablty saplg desg uder the above set up by drawg saples fro "sythetc populato". We have used ths estator o a populato of coutres (CO Sar-dal Swesso Wrete 99) for estatg the varace of ports aog coutres ad have show that the proposed estator s ore effcet as copared to ts copettors ters of relatve effcecy (RE)... Notatos We deote populato by ; ad the populato of ordered pars by U' { U : U ( U U) U U are uts tae fro U so that < }. The value of the study varable y o U be deoted by y ad o U be deoted

67 STATISTICS IN TRANSITION-ew seres Deceber by ( y y ). Defe for a real syetrc fucto h o ( y y ) t h( y y) for soe U T U N t N N ( ) N. T U for dfferet h fuctos:. For ( ) t( y) y y ( ) we get T U ( y y) Sy : N U fte populato varace.. For t( y z) ( y y)( z z) Cyz we get T U ( y y)( z z): populato covarace. N U N( N ) y y π π. For t( y) ( ππ π) we get T U y y ( ππ π) V < U π π G whch s ates ad Grudy-type varace of y Horvtz Thopso (HT) estator of a populato total HT π. For t y y gves T U y < U G-coeffcet. Note that () the fte populato varace of t values over the populato U. y y G whch s populato N( N ) y s S y has bee expressed as ea. Estato uder uequal probablty desg Assocated wth U let x be the sze easure so that y x ; P x x x x. Note for the fucto h () s such that t( y) t( x ) ; U t ( ) ( ) x x x wheever y x. Let the sze easures o U be t ( x ) so t ( x ) that P ( U ). We cosder here a probablty proportoal to sze t ( x ) U desg wth replaceet wth saple sze (s). Let the correspodg saple space be deoted by S. Deote ths desg by D ( U S P ). Based o ppswr saple s U t : U s the of sze ad the correspodg data d {( ) } ubased estator of TU ( Sy ) s gve by

68 8 M. K. Srvastava N. Srvastava H. P. Sgh: Full forato The varace of T s Ts s s gve by t N P (.) ( ) V T t tl P P N P N Pl s l U (.) Where s suato over all dfferet pars of -tuples U. U Furtherore the estator of ( s ) v T Where the suato s gve by t tl ( ) s NP NP l s (.) s over all pars of -tuples the saple s. Let us ow cosder a reducg trasforato o s that gores order ad ultplcty of -tuples ad results to a saple s ad the correspodg data be deoted by d* {( U t ): U s }. Sce the saple s s a suffcet statstc (Cassel Sardal ad Wreta 976) therefore by usg Rao- * Blacwellzato we get a proved ubased estator T s S s * * S * S S S E Ts D d E T s E E {}. T u wth varace { } sce { } * V ( T ) s S ˆ { { }} S S s E E T D * d * T u ˆ P ( s ) T P ( s ) T s ( s ) s S s.t. s gves s P s P s * V T V T Where ( ) ( ) s S s.t. s gves s s. u P further { s } { s } Whle utlzg forato o already selected -tuples a ppswr saple we get soe addtoal forato o few ore -tuples. For exaple f U ( U U) ad U ( U U) are selected a saple observg these - tuples also eas observg U U ad U. So we have effectvely derved forato o U addto to forato o U ad U where U ( U U). We expect that ths forato o U could also be utlzed to.

69 STATISTICS IN TRANSITION-ew seres Deceber get a better estator though t was ot drectly selected. Let the saple cotag forato o U U U be called full-forato saple s ad subset of dstct uts be deoted by deoted by ( ) { } { } A U U U U U U (as the above exaple). A could be derved fro ppswr saple s or by s (wth ther usual otatos) by a reducg fucto o s whch gves oly dstct uts appeared o -tuples s ( s ). The probablty dstrbuto of such a rado set A has bee stated P s P U. P U... P U. Defe a rado the theore.. We have ( ) ( ) ( ) ( ) set A U U U wth a eag that A s a set of dstct uts U 's appearg oe or ore U. Clearly A s a rado set the laguage of probablty theory tag ts values fro U as a subset. Let us deote the uber of dstct uts A by A ths uber s also a teger-valued rado varable. To dscuss the statstcal propertes of the estator based o A we would be terested the probablty dstrbuto of the rado set A. Cosder a set B of M uts beg fxed fro U as a subset aely U U U ; geerate a set of M ( ) l l l M M -tuples purely ade of uts B deote t by UB B M M N clearly UB U. If oe draws probablty proportoal to sze saples of sze fro U B wth replaceet (ppswr) the correspodg saple space be gve by S B. Note that S B s a collecto of all -tuples ade up etrely of the uts (labels) the subset B so that S S. The probablty of ay such saple ( B) { U U U } S the desg D ( s ) each B U s soe -tuple ade up of uts B uder ( B P ) ( s ) ( ). ( )... ( ) P U P U P U where U 's are ot ecessarly dstct. The probablty dstrbuto of the set A s gve by : Theore.. Cosder a probablty proportoal to sze saple s of sze draw fro U wth replaceet uder the ppswr desg D ( U S P ). Let the costtuet -tuples Defe a set valued rado varable tag ts values fro U s be deoted by U U... U.

70 0 M. K. Srvastava N. Srvastava H. P. Sgh: Full forato A U U... U as the set of dstct uts coo to U s. The sze of rage fro gve by to { N} for every B of sze. Next A.e. A would. The probablty dstrbuto of A s ( B ) ( ) A B ( ) P A P s B S B ( B) ( ) ( ) ( ) P A A B P s P A { A : S B A B A B} for every B of sze P A the secod ter o the rght had sde of the above expresso coes fro the prevous step. Proceedg slarly for for every B of sze dstrbuto of Appedx). where ( ) ( B) ( ) ( ) ( ) P A A B P s P A { A : ( ) S B A B A B}. Proceedg slarly up to oe gets coplete A recursvely. (proof of the theore has bee gve It ca easly be see that paraeter vector t ( ) statstc. Sce T s * ad T s t... t N ( ) P s P or ( ) P s s ( ) P s ( ) therefore ( s ) are both ubased estators of U or depedet of the A s a suffcet T ad s ( ) or A s suffcet therefore Rao Blawellzato of both the estators gves the sae ubased estator T defed as ( ) s ( f ) full ( s ) S A ( ) { } A P T T. E T s fullf s S s A P (.) where s over all such ppswr saples of sze s S that result to sae A gve A U S A where ppswr saples s reduces to a gve A.

71 STATISTICS IN TRANSITION-ew seres Deceber 009 S the above expectato s a rado varable tag ts values fro the subspace Ths estator s ubased sce S A f ( ) { } E A T A S A full E E T s T A U where E s over all the subsets s of U. Further the varace of the full-forato estator s gve by where s S ( ) { } V T A S A f E E T T s full s U P ( s ) T P ( A ) T A s so that ( ) U S A s gves A s ( ) s. U P s over all such ppswr saples S that gves the sae whch s beg fxed at the prevous suato. Here P ( A ) A A derved The- obtaed as the probablty dstrbuto of the rado sets ore.. Fally we get the followg relatoshp * ( ) fullf { s } { s} U A are V T V T V T (.5) s. Estato uder equal probablty desg Let the desg D ( U S P ) be srswr. We draw a sple rado saple s of sze fro U wth replaceet (srswr) ad deote the correspodg data by {( U ):U } d t s. Based o d we have a ubased estator of S y T s t f t s where f s the uber of tes the ad be the uber of dstct -tuples (.) th -tuple s repeated the saple s N N s. V ( T ) s srswr. S t

72 M. K. Srvastava N. Srvastava H. P. Sgh: Full forato t U U N where S ( t T ). Usg the result of Ra ad Khas (958) ad Des Ra (968)-pp0 theore.5 a better ubased (Rao-Blacwellzed) estator of s y based o saple s of dstct -tuples s s T E T t * S s s s s (.) N wth * V T s. S t V T s N We select -tuples fro U by srswor cosequetly we get a set A of dstct uts of U. These dstct uts allow us to ow the forato (tvalues) o dstct -tuples. For equal probablty wthout replaceet saplg the probablty dstrbuto of these uts s stated the followg corollary to theore.. I theore. we oted that the probabltes P ( A ) were all dfferet for dfferet A 's of the sae sze because these probabltes were depedg o the uts costtutg A whereas for equal probablty saplg the probabltes of sets A do ot deped o the uts costtutg A rather o the sze of A.e. A. Corollary.. Cosder the desg D ( U S P ) as sple rado saplg wthout replaceet. Cosder a srswor saple s of sze fro U uder the desg D the probablty dstrbuto of dstct uts A would be gve by N ( ) P ( ) N (.) ( ) P ( ) N P ( ) N N ( ) ( ) (.)

73 STATISTICS IN TRANSITION-ew seres Deceber 009 N P ( ) N for... where { :( ) } We propose a estator based o these ( ) ; ad { N} full-forato estator whch has bee defed as T ( ) s ( ) P ( ) N ( ) ( ) (.5) (.6) ( ) t dstct -tuples ad call t as (.7) The above proposed estator could be vewed as a Rao Blacwellzato of * the estator T s based o srswor of -tuples fro U gve the suffcet statstcs s that s T s( ) S ( ) s * ( S ) ( ) E T t. Moreover V T *. ( ) V s T s We shall ow state the expresso for varace of the proposed estator T s ad ts relatve effcecy the followg theore. ( ) ( ) Let us cosder the three codtog stages as follows: Stage I: Decde about the uber of dstct uts fro U. Stage II: Select a srswor of uts fro U. Gve these uts a total of -tuples are geerated. Stage III: A srswor of sze -tuples are selected fro the ( ) -tuples obtaed at stage II. Uder these codtog stages we shall prove the followg theore : Theore.. Uder the above stages of saple selecto of s the followg results hold :

74 M. K. Srvastava N. Srvastava H. P. Sgh: Full forato. The estator T s II s t II ad the proposed full- estate T U ubasedly for all values of. forato estator. V ( Ts II) V T s ( ) ( ) ( ) ( ) U { U U U } A t A t t t t t t ( ) A t U t ( ) ( ) ( ) Where A N N ( N ) A ; N ( N ) N ( N ) ( )( ) ; ( )( ) ( )( ) ( )( ) ( ) ( ) ad A N N N N N ( )( ). ( ) ( ) ( ) ( s ) ( ) ( ) U { U U U } V T E A B t E A B t t t t t t E ( ) A B t t C (say) Where A s are as () ad B s are ( ) defed as B U N ( ) ( ) ( ) ( )( ) B. N ( ) ( ) Ad. s( ) s( ) ; B ( ) ( )( ). ( ) ( ) N ( ) ( ) ( ) ( ) ( ) U { U U U } V T E V T E A t E A t t t t t t E ( ) A (say). U t t D T s over T s shall be gve by r ( ) 5. Relatve effcecy (RE) of usg ( ) RE C/D (.8) where C ad D are gve () ad () ad E correspods to the probablty dstrbuto of dstct uts the rado set A obtaed corollary.. (proof of the theore has bee gve Appedx). The calculatos of RE could easly be perfored by usg the probablty dstrbuto of obtaed corollary.. ;

75 STATISTICS IN TRANSITION-ew seres Deceber I case of sple rado saplg wthout replaceet desg D the fullforato estator T s becoes s y ( y ) y as a estator of ( ) s y whe t ( ) ( ) y y y ( ) N ( )... ; cyz ( y )( ) y z z of C yz y y < π π π ( ) where π π π (sce the preset desg D s srswor) a N N N whe vg ( ππ π ) ( ) estator of V G. A re-accoutg ad a ote o the applcato of the above results s requred at ths stage. I the theore. t has bee show that the full-forato estator s superor to the oe based o srswor saple Therefore T s( s T s ) ths suggests to draw dstct uts fro U to geerate ( ) -tuples to calculate T s ( ). Oe splest cosderato for decdg could be the cost fucto C C0 c. where C 0 beg the overhead cost ad c the cost of observg a ut U. Ths cost fucto s reasoable whe the aor cost s bor observg the uts ad the cost of calculatg the estate s coparatvely eglgble. Ths cosderato suggests us to select as large dstct uts fro C C0 U as pertted by. c To su up the above developet t s suggested that we shall tally select a saple of uts fro U. Next we cosder all -tuples geerated by t s ( ) ad estate T U by utlzg t-easures o geerated saple s ( ) of -tuples. For the suggested full forato estator we eed to ow the partto of S ppswr saple space o whch each saple reduces to a gve A ad also the codtoal probabltes o ths partto. Theore. gves the requred probablty dstrbuto of A the preset settg. Ths theore also helps detfyg s 's resultg to a fxed A startg wth ad progressg recursvely stead of checg every saple s aog N saples S oe by oe that gves the sae A.

76 6 M. K. Srvastava N. Srvastava H. P. Sgh: Full forato. Eprcal study The proposed estators were appled to estate the populato varace of ports o a real populato C0 of coutres cosstg of data o 98 port fgures ( llos of U.S. dollars) ad 98 fgures o gross atoal product ( tes of llos of U.S. dollars) (Sardal Swesso Wreta (99). The correlato betwee y(imp) ad x(gnp) ad t(y) ad t(x) are both hgh; 0.9 ad 0.9 respectvely. O the bass of coputer sulatos we have obtaed the followg results for the effceces of dscussed estators as copared to the estator T. s Table. Relatve Effceces Estator of y S T s * T s T s s y cov. ( ) Desg: (ppswr) Desg: (srswr) RE V ( T )/ V (). s I these calculatos the covetoal estator of s y has bee tae as s ycov by V ( s ycov ) N P ( y y ) whch s ubased for S s ( ) y. Its varace has bee gve ( y y) S U y. The results table show clearly that ( N ) P the proposed full forato estator clas overall superorty. Other estators of varace avalable lterature have ot bee cluded ths study sce these ad proposed estator are based o dfferet desgs therefore o coparable. Appedx PROOF OF THEOREM. Let the set B of M fxed uts fro U be gve by B { Ul U... } l U l M U l 's beg tae fro U so that M N. The ppswr saple s gves the rado set A { U U... } U A M. Uder the desg D

77 STATISTICS IN TRANSITION-ew seres Deceber The set ( ) ( B ) [ ] P A B P s (A.) S B A s a rado set whch s oe set aog the class of all subsets of U. The sze of A.e. A would rage fro Further to { N}. [ ] ( ) ( ) (A.) P A B P A P A { A : A B} { A : A O coparg () ad (5) we get A B} B P ( A ) P s (A.) { A : A S B A B} for every set B so that B M. If we solve (6) for P ( A ) ( M ) ( ) ( ) startg wth every B of sze.e. B oe gets probabltes of every sets A of sze. Next for soe fxed B of sze ( ) we have ( B P ) ( A ) A ( ) ( ) B P s P A B (A.) S B { A : A r A B} where the secod ter o rght had expresso coes fro probabltes P ( A ) at the prevous step. The probabltes of all such A 's each of sze ( ) could be calculated sply by tag varato over all subsets B of sze. N we get a coplete probablty Proceedg slarly for to { } dstrbuto of A. More geerally for ad for fxed B B ( B P ) ( A ) A ( ) ( ) B P s P A B S B (A.5)... { A : A A B} Varyg B each of sze ( ) oe gets the probabltes P ( A ) so that A B. Note that the secod ter o the rght sde of the above expresso coes fro probabltes P ( A ) obtaed at the prevous step. The above recursve ethod eables oe to get the coplete probablty dstrbuto of A. Ths copletes the proof.

78 8 M. K. Srvastava N. Srvastava H. P. Sgh: Full forato The advatage of the recursve procedure explaed Theore. s that oe eeds oly to calculate the probabltes of all such ppswr saples of sze ( M s ) s draw fro U M the space of all -tuples ade of uts fro fxed set B ad proceed wth M. For varatos B of sze oe gets P ( A ) wth B A. These probabltes are the used to calculate P'(A ") A for varatos B B A so that B. Proceedg slarly oe gets the coplete probablty dstrbuto of A. ILLUSTRATION OF THEOREM.: Cosder a ppswr desg ( ) U U U U U ; ad S cossts of 6 ppswr saples of sze DU S P { } ; the szes of the uts U U U U are respectvely gvg ther selecto probabltes ad 0. respectvely. Let us ow costruct a desg D ( U S P ) so that ( ) ( ) ( ) ( ) U { U U U U U U U U U U U U U ( U U ) ( U U ) U5 U6 }; S s saple space of ppswr saples of sze havg ( N ) 6 6 saples N where ( ) 6 N ; ad the duced probabltes are P ( ) 0.07 P P ( U ) P P ( U ) 0. P P ( U ) 0.6 P P ( U 5 ) 0. P 5 P ( U 6 ) 0.7 P 6. Note ad {r N} {6 }. Now startg wth M for the set B { } the theore gves ( ) ( ) ( ) ( ) ( B ) ( { } ) { } P A P s P { P ( U )} P.05 0 slarly we get P( A { } ) P.5 0 ; P( A { } ) P ; P( A { } ) P.7 0 ; P( A { } ) P5 0.06; P( A { } ) P Next cosder sets B of sze we get the probabltes of sets A of sze. Let us fx B { } we have U B {( ) ( ) ( )}; saple space uder ppswr fro U B would be gve by S B { U U U }{ U U U } { U U U } { U U U } { U U U } { U U U }{ U U U }{ U U U }{ U U U } all wth three U

79 STATISTICS IN TRANSITION-ew seres Deceber ( B ) S P ( s ) becoes perutatos ad { U U U } wth perutatos. B P P P { P ( P P ) P ( P P ) P ( P P )}! ( P. P. P ). The result of the theore gves ( B ({ ) }) S B ( ) ( ) (A.6) P P s P A { A : A A B} Note P ( A ) A have bee calculated at the prevous step. Ths gves ({ }) ( ) ( ) ( ) { } ( ) P P P P P P P P P P 6 P. P. P slarly P ({ } ) P ({ } ) 0.6 P ({ } ) Havg obtaed the probabltes of sets A of sze we would ow obta A { } fx B { }. Note that SB S sce A U therefore ( B ) S P ( s ). The result of the theore gves B ( B ) ({ }) ( ) ( ) ( ) P P s P A P A S B { A : A { A : A A B} A B} PROOF OF COROLLAR.. Choose ad fx M uts fro U ad deote the by a subset B B M M N ad draw a srswor saple s of sze fro U P ( s ) ; the correspodg saple space s deoted by S N ad the set of dstct uts s be deoted by A { U U... U } wth sze A where U U... U gves dstct uts whch have appeared U 's s. I U there are M -tuples cosstg etrely of the uts B we deote the by U U U. The M M [ ] P A B M ( ) N ( ) (A.7)

80 0 M. K. Srvastava N. Srvastava H. P. Sgh: Full forato sce r-tuples ways N ( ) s could be chose ( M ) M where M ( ) ( ) ways fro U M out of total N N B M ad A M. Let the probabltes of A havg dstct uts be deoted by P ( ) for.... We shall observe that P ( A ) for A U depeds oly o the sze of A.e. A ot o whch labels costtute A. I other words the dstrbuto of A s varat uder the perutatos of U s. Thus for soe U such that A we have A P P ( ) ( A ) (A.8) But [ ] ( ) N ( ) P A B P T B M { TT : B} M ( ) P ( A ) Fro () we get P [ A M B] ( ) A P ( ) (A.9) O coparg (0) ad () we get M ( ) N ( ) M ( ) N ( ) N ( ) P ( ) M (A.0) These equatos ca be recursvely solved for P ( ) startg wth M we get P ( ) ( ) ( ) N ( ) N ( ) ( ) P ( ) P ( N ) N N ( ) ( )

81 STATISTICS IN TRANSITION-ew seres Deceber 009 ( ) r P N ( ) N r ( ) P ( ) N ( ) ( ) For.... Ths s the desred dstrbuto of uber of dstct uts a srswor s tae fro U uder the desg D ( U S P ). ILLUSTRATION OF COROLLAR. Let N 5. The ad 5 ad the equato (7) ad (8) reduces to 5 5 P ; ( ) P P ( ) ( ) ( ) ( ) ad P ( ) P ( ) P ( ) 5. Thus ( ) ( ) P ; P..5 ad P ( 5) 6 0 PROOF OF THEOREM.. The codtoal expectato of T s gve ad II s E ( T II) ad EE ( Ts II ) ET s ( ) ( ) ( ) ( ) ( ) ( ) ( ) t ( ) T s ( ) ( U ( )) E t t P U U U. U t T U N ( N ) ( ) ( ) ( ) of. Ths shows that T s ad T s ( ) are ubased estators of T U for all values

82 M. K. Srvastava N. Srvastava H. P. Sgh: Full forato. Now V ( T II) V E ( II) E V ( II) V E ( II) ( ) V T s Showg Here where ( ) T s ( ) s s superor to T II. Let us ow cosder s ( ) ( ) V T E t ( ) E t s {( )} ( ) E t calculates to be ( ) ( )( ) ( ) ( ) ( ) t tt tt tt N ( N ) U N ( N )( N ) U U U U ( ) () l < < l l ; ( ) ( )( )( ) ( )( )( ) N N N N ( ) U ( ) ( ) l < < l tt U ; ( ) ( ) ( ) ( ) ; U ad E t ( ) calculates to be ( ) l < < < l N ( N ) { t ( ) ( ) t t t t U U U ( ) U t t (A.) U ( ) U t t }. ( ) ( ) < < O puttg these values () we get ( ) ( ) ( ) ( ) ( ) { } V T s A U t A U t t U t t U t t A U t t ( ) ( ) ( ) Where A ; N N ( )( ) ( N ) A ; N ( )( ) ( N ) N ( N ) ( )( ) ( ) A ; N ( N )( N ) N ( N ) ( )( ) ;

83 STATISTICS IN TRANSITION-ew seres Deceber 009. Next cosder ( ) ( ) V Ts t t ( ) ( ) ( ) Tag E o V T s soe algebrac splfcato gves ( ) ( ) ( ) ( ) { } E V T B U t B U t t U t t U t t B U t t s (A.) Where ; ( ) N B ( ) ( ) ( ) ( ) N ( )( ) ( ) B... ( ) ( ) N ( )( ) B.. ; So far we have bee treatg as a quatty fxed advace but actual N -tuples by srswor. For ay practce we fx ad select -tuples fro ( ) gve saple s we get dstct uts of U. Thus s actually a rado varable. The dstrbuto of has already bee calculated corollary.. by recursve ethod. We ow have ( s ) () () () V T E E V E V E V E E ( ) ( ) ( ) ( ) ( ) U { U U U } E A B t E A B t t t t t t. Fro (5) we get ( ) E A B t t C (say) ( ) (A.) U V T s ( ) E V T s ( ) ( ) ( ) ( ) ( ) ( ) U { U U U } E A t E A t t t t t t ( ) E A t t D (say) ( ) (A.) U

84 M. K. Srvastava N. Srvastava H. P. Sgh: Full forato 5. The relatve effcecy (RE) of the estator RE ( s ) V T V T s ( ) C D T s ( ) has bee gve by REFERENCES [] CASSEL C.M. SARNDAL C.E. ad WRETMAN J.H. (977). Foudatos of Iferece Survey Saplg. New or: Wley. [] CHAUDHRI A.(978). O Estatg the Varace of a Fte Populato. Metra [] DATTA G.S. ad GHOSE M.(99).Bayesa Estato of Fte Populato Varaces wth Auxlary Iforato. Sahya Ser. B [] DATTA ad TIWARI (99).Bayesa Estato of Fte Populato Varaces wth Auxlary Iforato. Sahya Ser. B [5] GHOSH M. ad LAHIRI P.(987).Robust Eprcal Bayes Estato of Varace Fro Stratfed Saples..]. Aer. Statst Assoc [6] GHOSE M. ad MEEDEN G.(98).Estato of the Varace Fte Populato Saplg. Sahya Ser. B [7] GHOSE M. ad MEEDEN G.(98).A ew Bayesa aalyss of a rado effect odel. J.R. Statst. Soc.B [8] ERICSON W.A. (969).Subectve Bayesa Models Saplg Fte Populatos(Wth dscusso). Joural of Royal Statstcal Socety Ser. B 95. [9] GODAMBE V.P.(955). A ufed theory of saplg fro fte populatos. J. R. Statst. Soc. B [0] HANURAV T.V.(966). Soe aspect of ufed saplg theory. Sahya Ser. A [] LAHIRI P. ad TIWARI R.C.(99). NoParaetrc Bayes ad Eprcal Bayes Estato of Varaces fro Stratfed Saplg. Sahya Ser. S

85 STATISTICS IN TRANSITION-ew seres Deceber [] LIU T.P. (97a). Bayes Estato for the Varace of a fte populato. Metra 6 part I. [] LIU T.P. (97b). A Geeral Ubased Estator for the Varace of a Fte Populato. Sahya Ser. C 6 part I. [] LIU T. P. ad THOMPSON M.E. (98). Propertes of Estators of Quadratc Fte Populato Fucto the Batch Approach. Aals of Statstcs [5] MUKHOPADHA P.(978). Estatg the Varace of a Fte Populato Uder a Superpopulato Model. Metra 5 5. [6] MUKHOPADHA P. ad BHATTACHARA S. (989). O Estatg the Varace of a Fte Populato Uder a Superpopulato Model. Joural of the Ida Statstcal Assoc [7] RAJ DES (968). Saplg theory. Tata McGraw-Hll Publshg Copay Ltd. New Delh. [8] SWAIN A.K.P.C. ad MISHRA G.(99). Estato of Fte Populato Varace Uder Uequal Probablty Saplg. Sahya Ser. B [9] SARNDAL.C.E. SWENSSON B. ad WRETMAN J. (99).Model Asssted Survey Saplg Sprger-VerlagNewyor. [0] VARDEMAN S. ad MEEDEN G. (98). Adssble Estators Fte Populato Saplg Eployg Varous Types of Pror Iforato. Joural of Statstcal Plag Iferece 7 9. [] WUC. ad SITTER R.R. (00). Effcet Estato of Quadratc Fte Populato Fuctos the Presece of Auxlary Iforato. J. Aer. Statst. Assoc [] ZACKS S. ad SOLOMAN H. (l98). Bayes ad Equvarat Estators of the Varace of a fte populato:part I Sple rado Saplg. Cou. Statst-Theory Meth. A

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87 STATISTICS IN TRANSITION-ew seres Deceber STATISTICS IN TRANSITION-ew seres Deceber 009 Vol. 0 No. pp. 7 A NONPARAMETRIC CONFIDENCE INTERVAL FOR AT-RISK-OF-POVERT-RATE Wocech Zelńs ABSTRACT I the Europea Cosso Eurostat docuet Doc. IPSE/65/0/EN page the ``at-rs-of-poverty rate" (ARPR) s defed as a percet of populato wth coe saller tha 60% of populato eda. Zelńs (008) proposed a dstrbuto-free cofdece terval for ARPR. I the paper a exaple of applcato of the costructed cofdece terval s show. Key words: boal dstrbuto cofdece terval ARPR.. Itroducto I the Europea Cosso Eurostat docuet Doc. IPSE/65/0/EN page the ``at-rs-of-poverty rate" (ARPR) s defed as follows. Let EQ_INC deote the equvalsed dsposable coe of the -th perso ad let weght deote the weght of perso. The ``at-rs-of-poverty threshold" (ARPT) s calculated as 60% of calculated eda value.e. where ARPT At rs of poverty threshold 60%EQ_INC MEDIAN W ( EQ _ INC EQ _ INC ) f weght EQ _ INCMEDIAN W EQ _ INC f weght < < weght ad W All persos weght. Departet of Ecooetrcs ad Statstcs Warsaw Uversty of Lfe Sceces Nowoursyowsa Warszawa e-al: [email protected].

88 8 W. Zelńs: A oparaetrc cofdece The the ``at-rs-of-poverty rate" s calculated as the percetage of persos (over the total populato) wth a equvalsed dsposable coe below the atrs-of-poverty threshold (.e. the equvalsed dsposable coe of each perso s copared wth at-rs-of-poverty threshold). The cuulated weghts of persos whose equvalsed dsposable coe s below the at-rs-of-poverty threshold s dvded by the cuulated weghts of the total populato (.e. su of all the persoal weghts): weght All persos wth EQ _ INC< atrs of poverty threshold ARPR 00%. W The atural estator of ARPR s as follows. Let be a saple of dsposable coes of radoly draw persos ad Med deotes the saple eda. The estator ARPR s defed: ARPR #{ 0.6 Med } where #S deotes the cardalty of the set S. The propertes of the above estator (bas varace etc.) depeds strogly o the dstrbuto F of populato coe. Zelńs (006) showed that the estator s alost ubased.e. E ARPR ARPR F(0.6 Q(0.5)) F for all cotuous F (Q(0.5) stads for the eda of the dstrbuto F). He also calculated ts varace. However the proble s terval estato. Zelńs (007) proposed a oparaetrc cofdece terval for ARPR. It appeared that hs cofdece terval s too coservatve.e. the true cofdece level s sgfcatly larger that the oal oe. I what follows we costruct a cofdece terval for ARPR the cofdece level of whch s ear oal oe.. Cofdece terval Let F deotes the cdf of a dstrbuto of populato coe. It s assued that F s cotuous. We are terested estato of the paraeter θ F( αq( q)) for gve α q (0) where Q ( ) deotes the quatle fucto ( Qx ( ) F ( x)). For α 0.6 ad q 0.5 paraeter θ s ARPR. We are terested costructg a cofdece terval for θ.

89 STATISTICS IN TRANSITION-ew seres Deceber Let be a saple fro F ad let : < < : be order statstcs. As a estator of θ we tae ˆ θ #{ α M: } where M dqt ( dt a s the greatest teger ot greater tha a). Here M: s a estator of q-quatle Q(q) of the F dstrbuto. Let ξ be the uber of observatos ot greater tha α M: : The dstrbuto of ξ s ξ #{ α }. M: P { ξ } P { ξ } P { ξ } P { α } F F F F : M: } P { α } 0 M. F : M: where (Davd ad Nagaraa 00)! PF{ : α M: } ( F( v)) ( )!( M )!( M)! M f ( v) F( u) [ F( v) F( u)] f ( u) dudv! M! M M ( v) u [ v u] dudv 0 0 ( )!( M )!( M)! )( ) F( αq( v)) B 0 M bm M () v dv v Here B ab ( ) ad b ab ( ) deotes cdf ad pdf of beta dstrbuto wth paraeters (ab) respectvely. That dstrbuto ay be wrtte the for F( αqq ( )) PF{ : α M: } B M q F( αq( v)) F( αq( q)) B 0 M BM bm M () v dv. v q

90 0 W. Zelńs: A oparaetrc cofdece It s well ow that f S s a rado varable dstrbuted as boal wth paraeters ad p the Pp { S } p ( p) B ( p). 0 Hece the dstrbuto of ξ s alost boal wth paraeters M- ad F( αqq ( )). q If F s power dstrbuto wth shape paraeter b.e. the ad 0 F( x) x b 0< x< F( αq( v)) b α v F( αq( v)) F( αq( q)) BM BM bm M () v dv 0. v q Hece case of power fucto dstrbuto ξ s boally dstrbuted wth paraeters M- ad α b. For ths dstrbuto θ α b q. Let γ (0). Cosder a terval (see Appedx) γ γ ξ ξ ; ; ξ ξ; qb M qb M where B - (ab;δ) s the δ quatle of beta dstrbuto wth paraeters (ab). If F s power fucto dstrbuto the γ γ θ ξ ξ ; ; ξ ξ; γ. PF qb M qb M Hece (*) s a cofdece terval for θ. The questo s: what s the value of γ γ θ ξ ξ ; ; ξ ξ; PF qb M qb M (*) (**) for dstrbutos F other the power fucto oe? I geeral t s possble to calculate (**) because t strogly depeds o F. I tables - there are calculated

91 STATISTICS IN TRANSITION-ew seres Deceber 009 probabltes (**) (deoted by ˆ γ ad the ea legth ( ˆΔ ) of cofdece terval (*) for: b Pareto dstrbuto wth pdf b x b Gaa dstrbuto wth pdf x e Γ( b) for x (0 ) ; x for x (0 ) ; lx Logoral dstrbuto wth pdf exp for x (0 ). xb π b Calculatos were ade for α0.6 q M d/t ad γ Coclusos The cofdece terval for θf(α Q(q)) ay be costructed as a cofdece terval boal dstrbuto based o the uber of observatos ot greater tha α M:. Its cofdece level for power fucto dstrbuto s exactly the sae as the cofdece level of cofdece terval for boal proporto. For Pareto Gaa ad Logoral dstrbutos cofdece level does ot dffer uch fro the oal oe. It ay be expected that for other cotuous dstrbutos the cofdece level wll behave slarly. Appedx: cofdece terval for boal proporto Let η be a boal rado varable wth paraeters ad uow p. It s well ow that P { η } B ( p) ad P { ξ } B ( p). p p Let δ (0) be a gve uber. Cofdece terval for p at the cofdece level δ s defed as P { p ( η) p p ( η)} δ for all p (0). p L U For gve ad let p L () be the soluto of δ δ B ( pl( )) or equvaletly B ( pl( )). We obta pl( ) B ;. δ

92 W. Zelńs: A oparaetrc cofdece Slarly we obta pu ( ) B ;. Hece the cofdece terval for p at the cofdece level δ s of the for δ ; δ δ η η η η; δ for all (0). Pp B p B p The actual cofdece level s hgher tha the oal oe because of dscreteess of boal dstrbuto (see for exaple Brow et al. 00). Table. Pareto dstrbuto b ARPR ˆ γ ˆΔ ˆ γ ˆΔ ˆ γ ˆΔ

93 STATISTICS IN TRANSITION-ew seres Deceber 009 Table. Gaa dstrbuto b ARPR ˆ γ ˆΔ ˆ γ ˆΔ ˆ γ ˆΔ

94 W. Zelńs: A oparaetrc cofdece Table. Logoral dstrbuto b ARPR ˆ γ ˆΔ ˆ γ ˆΔ ˆ γ ˆΔ REFERENCES BROWN L. D CAI T. T DASGUPTA A. (00) Iterval estato for Boal proporto Statstcal Scece 6 0. DAVID H. A. NAGARAJA H. N. (00) Order Statstcs Thrd Edto Wley. ZIELIŃSKI R. (006) Exact dstrbuto of the atural ARPR estator sall saples fro fte populatos Statstcs I Trasto ZIELIŃSKI R. (007) A cofdece terval for ARPR "at-rs-of-povertyrate" Statstcs I Trasto 8 7.

95 STATISTICS IN TRANSITION-ew seres Deceber STATISTICS IN TRANSITION-ew seres Deceber 009 Vol. 0 No. pp KURTOSIS OF A RANDOM VECTOR SPECIAL TPES OF DISTIBUTIONS Katarzya Budy Ja Tatar ABSTRACT I preseted paper the authors attept to geeralze defto of urtoss for the case of ultdesoal ad prove ts essetal propertes. The geeralzed characterstc appled the sgle-deso case has the sae propertes as urtoss that s ow the lterature o sgle-desoal rado varables. The bass of coducted cosderatos s the defto of the power of a vector space wth the scalar product. Key words: Kurtoss Moets of a rado vector Multdesoal dstrbuto Power of a vector.. Itroducto Ths paper s a cotuato of prevous studes by Tatar ( a 000b00 00) Osewals ad Tatar (999). Let us recoll: Let there be ay vector space ( R R ) wth the classcal (Eucldea) scalar product for v ( v v... v ) w ( w w... w ) R : v w v R ad ay uber {} v w. For ay N 0 N 0 a followg defto of -th power of the vector v has bee proposed. Defto ad v o R Cracow Uversty of Ecoocs Departet of Matheatcs [email protected]. Cracow Uversty of Ecoocs Departet of Matheatcs [email protected].

96 6 K. Budy J. Tatar: Kurtoss of a rado v v v v v for odd for eve Fro the above-etoed defto edately result two followg portat propertes N0 : v R N : v R eve odd v v R R. Defto For ay uber N 0 a ordary oet of the order of a rado g ξ ξ t eas vector ξ we call the expected value of a rado varables ( ) E( ξ ) f E( ξ ) < Partcularly for / 0. ad for ay N we have () ( ) / ( ) 0 0 / for - eve for odd The sybols le used the forula () eas the classcal xed ordary oets of the ra. Slarly to the ordary oets we defe cetral oets of ultdesoal dstrbuto. We oly eed to assue the defto that g ( ξ ) ( ξ E(ξ )). Fro the defto ad results ter ala the followg geeral cocluso: N N 0 : - eve R : - odd R. I prevously cted papers o the bass of defto of the power of a vector ad the ordary oet of a rado vector portat results for characterzato of ultdesoal probablty dstrbutos were obtaed. Iter ala Chebyshev s Iequalty ow for sgle-deso case was geeralzed ultdesoal versos of the law of large ubers were proposed there were defed ad appled tools whch help to aalyze ad easure the asyetry of

97 STATISTICS IN TRANSITION-ew seres Deceber ultdesoal probablty dstrbutos the defto of covarace ad correlato coeffcet was geeralzed as well. We would le to ephasze that proposed ad developed cocepto essetally ad drectly provdes us wth characterstcs of aalyzed rado vector ot properly costructed (sgledesoal) fuctos of ts ters. To exercse our prevous declarato let s ove to the proposto of easureet of cocetrato (whch eas flatteg as well) probablty dstrbuto of a rado vector.. Propostos At the begg let s recollect that the theory of sgle-desoal rado varables capacty of cocetrato dstrbuto easure aroud expected value t s assued ter ala ts fourth cetral oet. Usually order to obta relatve easure the fourth order cetral oet s furtherore dvded by square varace. The hgh values of dcator costructed ths way (urtoss) dcate that there s the sgfcat tedecy to focus the dstrbuto aroud ts expected value ad vce versa: dcators low level shows ts flatteg. The defto of rado vector cetral oet let us to defte the urtoss a slar way for the ultdesoal dstrbuto. Defto... : Ω R the followg The urtoss of a rado vector ( ) value wll be called: Kurt [( E ) ] E. () ( D ) I the proofs of theores ad whch we ll for the further part of ths paper we ll use the followg lea. Lea Let (... ): Ω R be a rado vector fulfllg codtos: a) ~ N ( σ ) for ay {...} b) for all {... } : f tha ad are stochastcally depedet (sybolcally: ). The we have the followg equaltes:. E( ) ( )( σ ) ( σ 6 σ ) σ

98 8 K. Budy J. Tatar: Kurtoss of a rado. E( E ) ( ) ( σ ) σ. ( E ) ( ) E σ. E E( ) ( ) 5. E σ E E. Proof: Let s recollect that f a rado varable ξ : Ω R has a oral dstrbuto wth a ea of ad a varace of ξ~n σ the E σ that s ( ) ( ξ ) σ E ( ξ ) σ ad ( ) E ξ σ 6 σ. () Whle usg the propertes of tegral (alteratvely su) ad the depedece of rado varables ad... such that we get for all { } ( ) E E E( ) ( ) E( ) E( ) E. () Thus usg () the equalty () taes for: E ( ) ( )( σ ) ( σ 6 σ ) σ Ad that eds the proof for thess. Successvely fro the depedece of rado varables ad ad for () of the ordary oets of the secod ad thrd order a rado varable wth oral dstrbuto we get thess.

99 STATISTICS IN TRANSITION-ew seres Deceber I fact: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ). E E E E E E E σ σ σ σ I order to prove the thess we use the depedece of rado varables ad as well. So there s: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ). E E E E E E E E E E E σ Fally for thess we oly eed to coduct the followg reasog: ( ) ( ) E E E E E ( ) ( ) ( ) ( ) σ σ σ. Thess 5 s ths way obvously fulflled. Proof of the lea has bee fshed. At the oet the theore cocerg the urtoss of a rado vector wth oral argal dstrbutos wll be defed ad proved. Theore If ( ) R Ω :... s a rado vector fulfllg lea assuptos the Kurt σ σ σ ( ) D D D. (5)

100 50 K. Budy J. Tatar: Kurtoss of a rado Proof: Usg the defto of eve power of the vector propertes of scalar product ad tegral (su) we get: E ( E ) E E E E [( E E E ) ] E [ ] E E E E E E E E. Let s otce that we also have equalty: Ideed: E ( E E E ) ( E E E ) ( E E ) E E E (6) E E E ( E ) E ( E E ). Tag to cosderato the equalty (6) the fourth cetral oet ca be show the followg way [( E ) ] E( ) E( E ) E( E ) E (7) E E E ( ) E E. Successvely fro the lea equalty (7) taes for: Ideed: E [( E ) ] σ σ σ. (8)

101 STATISTICS IN TRANSITION-ew seres Deceber ( ) [ ] ( )( ) ( ) 6 E E σ σ σ σ ( ) ( ) ( ) σ σ σ ( ) σ ( ) ( ) σ σ σ σ σ σ σ σ σ ( ) ( ) σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ 8 σ σ σ σ 8 σ σ σ σ σ

102 5 K. Budy J. Tatar: Kurtoss of a rado σ σ σ σ σ σ. Furtherore the followg sequece of equaltes s true: ( D ) D σ σ σ. (9) I the vew of (8) (9) ad fro defto urtoss of a rado vector fulfllg lea assuptos ca be expressed as: Kurt σ σ σ Ad that fshes the proof of theore. It s easy to otce that sple ad edate cosequece of theore s the followg cocluso. Cocluso If (... ): Ω R s a rado vector fulfllg assuptos of lea ad f there s also codto σ σ... σ σ the Kurt. Fro the cocluso above also coes out qute obvous but t let us to call the subtted proposto as geeralzato a rear that urtoss of a sgledesoal rado varable orally dstrbuted equals. I order to realze the purpose forulated at the begg of ths paper the ext theore we ll get the for of urtoss of a rado vector whch argal varables have a Studet's t-dstrbutos. Theore Let (... ): Ω R be a rado vector fulfllg followg codtos: a) ~ t ν (that s has a Studet's t-dstrbuto wth ν degrees of freedo) where ν > for each {...}.

103 STATISTICS IN TRANSITION-ew seres Deceber b) for all {... } The : f the ad depedet (sybolcally: ). 6 ν ν ν Kurt ν ν ν ν are stochastcally. (0) Proof: We ll coduct the reasog slar to the oe the proof for theore. But frst let s recollect that f rado varable ξ : Ω R has a Studet's t- dstrbuto wth v degrees of freedo (else: ξ ~ t ) where ν > the: ν ξ E( ξ ) D ξ E( ) 0 E ( ) 0 E ( ξ ) ν ν 6 ν ν ν ξ 6 6 ad Kurtξ Excess ξ. ν ν We also eed to otce that for vector whch was dscussed the assupto of theore we have the equalty E E E...E So here eerges cocluso that the ( ) ( ) thess of lea s equvalet for lea (wth the dfferece that ow we assue that the argal varables of a rado vector have a Studet's t- dstrbuto) taes for:. E( ). E ( E ) 0. E( E ) 0. E E E( ) 0 5. E E 0. The: ν ν ν ν 6 ν ν ν

104 5 K. Budy J. Tatar: Kurtoss of a rado ( ) E E ( ) E () 6 ν ν ν ν ν ν ν. ad ( ) D ν ν ν ν ν ν. () So fro () ad () ad fro defto we get: 6 Kurt ν ν ν ν ν ν ν ν ν ν ν 6 ν ν ν ν ν ν ν ν ν ν ν ν ν 6 ν ν ν ν ν ν ν ν ν ν ν 6 ν ν ν ν ν ν ν ad that s the requred thess (0). Also fro theore results the followg cocluso:

105 STATISTICS IN TRANSITION-ew seres Deceber Cocluso If (... ): Ω R s a rado vector fulfllg assuptos of theore ad f we have codto ν ν... v ν > as well the 6 Kurt ν. Proof: We oly eed to otce that the vew of equal uber of degrees of freedo of all argal varables the followg equaltes - fro theore are true: 6 ν 6 Kurt ν ν ν. ν ν Obvously for the urtoss of a rado varable wth a Studet's t-dstrbuto depeds oly o a uber of ts degrees of freedo ad equals 6 Kurt. v Let s otce that fro cocluso ad we obvously get: Cocluso If (... ): Ω R s a rado vector fulfllg assuptos of lea but σ σ... σ σ ad f (... ): Ω R s a rado vector fulfllg assuptos of theore ad we have codto ν ν... v ν > the Kurt > Kurt. REFERENCES FELLER W Wstęp do rachuu prawdopodobeństwa Volue ad. Warszawa: PWN. FISZ M Rachue prawdopodobeństwa statystya ateatycza. Warszawa: PWN.

106 56 K. Budy J. Tatar: Kurtoss of a rado JAKUBOWSKI J. SZTENCEL R. 00. Wstęp do rachuu prawdopodobeństwa. rd ed. Warszawa: Scrpt. OSIEWALSKI J. TATAR J Przegląd Statystyczy. Multvarate Chebyshev equalty based o a ew defto of oets of a rado vector. PLUCIŃSKA A. PLUCIŃSKI E Probablstya. Rachue prawdopodobeństwa statystya ateatycza procesy stochastycze. Warszawa: WNT. TATAR J Przegląd Statystyczy. O etórych arach rozproszea rozładów prawdopodobeństwa /. TATAR J Przegląd Statystyczy. Moets of a rado varable a Hlbert space. TATAR J. (000a) Nowa charateryzaca welowyarowych rozładów prawdopodobeństwa. Sprawozdae z badań statutowych; u. r: 9/KM//99/S; AE Kraów. TATAR J. 000b. Moety absolute welowyarowych rozładów prawdopodobeństwa. I: Polsa Aadea Nau Posedzee Kos Statystyczo-Deografcze PAN O/Kraów. Cracow 7 Noveber 000. TATAR J. 00. Nowe ary zależośc wetorów losowych. I: Polsa Aadea Nau Posedzee Kos Statystyczo-Deografcze PAN O/Kraów. Cracow May 00. TATAR J. 00. Prace auowe AE we Wrocławu. Prawa welch lczb dla welowyarowych wetorów losowych 006.

107 STATISTICS IN TRANSITION-ew seres Deceber STATISTICS IN TRANSITION-ew seres Deceber 009 Vol. 0 No. pp COMBINED EFFECT OF FAULT DETECTION AND FAULT INTRODUCTION RATE ON SOFTWARE RELIABILIT MODELLING S. Chatteree L.N. Upadhyaya J.B. Sgh ad S. Nga ABSTRACT Ths paper proposes a software relablty growth odel to study the cobed effect of creasg error detecto ad decreasg error troducto rate uder perfect debuggg. The odel s developed based o o hoogeeous Posso process (NHPP) ad ca be used to estate ad predct the relablty as well as the cost of a software product. Soe real lfe data has bee used to valdate the proposed odel ad to show ts usefuless. Coparso of ths odel wth other has bee carred out. Key words: Software Relablty Iperfect Debuggg Multple Falures No Hoogeeous Posso Process Icreasg Fault Detecto Rate Decreasg Fault Reoval Rate. Noeclature N (t) coutg process represetg the cuulatve uber of errors Po ((t)) Posso process wth ea (t) A ea tal error cotet the software b fault detecto rate per type fault p cotet proporto of type fault λ (t) fault detecto rate per ut te of type error λ(t) fault detecto rate per ut te (t) uber of errors to be evetually detected β error troducto rate (t) expected uber of type errors by te t. T total test te.e. release te s cuulatve te Assstat Professor Dept. of Appled Matheatcs ISMU Dhabad-8600 Correspodg author. Eal: [email protected]. Professor Dept. of Appled Matheatcs ISMU Dhabad Research Scholar Dept. of Appled Matheatcs ISMU Dhabad. Proect Fellow Departet of Appled Matheatcs ISMU Dhabad.

108 58 S. Chatteree L. N. Upadhyaya J. B. Sgh S. Nga: Cobed Effect R(x/t) codtoal relablty of software le axu lelhood estate Itroducto Coputers are wdely used varous felds of lfe cludg busess ad safety crtcal systes. Applcato of coputer eas the applcato of software. Therefore there exsts a creasg dead of hghly relable software. Sce assesset of software relablty s oe of the aor cocer preset-day software dustres oe eed a good atheatcal odel to estate the relablty of software. Research the area of software relablty has bee gog o sce last three decades. Detal studes related to software relablty has bee preseted []. Itally varous software relablty odels were developed usg the cocept of perfect debuggg process. Soe of the portat software relablty odels aog the are [567890]. Latter stage varous software relablty odels were developed usg the cocept of perfect debuggg. Soe of the also throws lght to other aspects of software relablty studes le: estato of cost release te etc.[567890]. Preset-day software developet process has becae very coplex. Therefore there s a eed to develop a effcet atheatcal odel whch ca gve better predcto of software relablty ad tae care of dfferet aspects of software developet process as well. Aog these aspects of software developet process error detecto rate.e. FDR ad error troducto rate.e. FIR durg software debuggg plays very crucal role the growth of software relablty. Pha[9] developed a software relablty odel cosderg perfect debuggg ad presece of ultple falures. He cosdered FDR ad FIR both as costat ad dfferet for dfferet types of errors. Also he cosdered presece of three types of errors: type errors.e. crtcal errors whch are very dffcult to detect type errors.e. aor errors whch are dffcult to detect type errors.e. or error whch are easy to detect. Software testg as well as debuggg are doe by hua beg. As te progress test persoel lears ore about the software. As a result FDR creases ad FIR decreases wth respect to te. I ths paper the FIR β ad FDR b are beg cosdered as a fucto of cuulatve te ad a ore realstc software relablty odel has bee developed to study the cobed effect of creasg FDR ad decreasg FIR o the growth of software relablty. As error troducto ad detecto depeds o the owledge of the test persoel these factors caot be dfferet for dfferet types of errors. Due to ths reaso FDR has bee cosdered the sae for all types of errors ad the sae s the case for FIR. Also ths paper presece of types of errors (for geeralzato) s cosdered. The proposed odel s based o o hoogeeous Posso process ad soe real lfe data are used to valdate the odel.

109 STATISTICS IN TRANSITION-ew seres Deceber Model developet A o hoogeeous Posso process based odel cosderg perfect debuggg ad presece of types of error has bee proposed ths secto. Model Assuptos The followg assuptos are ade for the developet of the odel. () Durg reoval of detected errors t s possble to troduce ew errors. () The probablty of fdg a error software s proportoal to the uber of reag errors the software. () There exst types of errors software. (v) The error detecto process follows a o hoogeeous Posso process. (v) Fault detecto rate (FDR) b s sae for types of errors ad creases wth respect to cuulatve te. Here b s cosdered as a logstc fucto.e. b where α 0 ad r are costat s s the cuulatve te. rs ( ) e α 0 (v) Fault troducto rate (FIR) β s sae for types of errors ad decreases wth respect to cuulatve te where β s For atheatcal splfcato the cuulatve te s s cosdered here. Otherwse aalyss wll be ore coplcated. Accordg to the assupto (v) the odel has bee forulated as Pr{ N ( t) } po( ( t)) 0... To obta the relablty of software the ea value fucto (t).e. the expected uber of software falures to be detered. It s obtaed by solvg the followg dfferetal equatos. d ( t) λ ( t) dt d ( t) b[ ( t) ( t)] dt d ( t) d ( t) β ( t) ( t) where ( 0) ap (0) 0 dt dt Solutos of the above dfferetal equatos gve the expresso for ( t) λ(t) ad R( x / t) as follows. ( t) ap [ e ( β ) ( β ) bt ] ( β ) bt λ( t) ap be ad

110 60 S. Chatteree L. N. Upadhyaya J. B. Sgh S. Nga: Cobed Effect bt ( x) e ( β ) [ ( t x) ( t)] R( x / t) e e Results & Dscusso To deostrate the usefuless of the proposed odel ad for coparso wth Pha odel [9] presece of three types of errors.e. a software s cosdered. Three types of errors are: type errors.e. crtcal errors whch are very dffcult to detect type errors.e. aor errors whch are dffcult to detect type errors.e. or error whch are easy to detect. For llustrato purpose the falure data of Msra [] gve Table-I are used. Here p 0.07 p 0. p The values of the costat r α 0 are.5 ad 0.05 respectvely. Table-I. Orgal falure data Test Wee Test Hours Crtcal errors Maor errors Mor errors Test Wee Test Hours Crtcal errors Maor errors Mor errors To estate the error a le s used. Solvg the le equato the estated value of errors obtaed s a ˆ. Whle Pha [9] estated the total uber of errors usg hs odel as a ˆ Though the estated total uber of

111 STATISTICS IN TRANSITION-ew seres Deceber errors a ˆ usg the proposed odel s less tha the actual errors preset the software stll t s better tha the result obtaed [9]. Better results ca be obtaed usg approprate fucto for b ad β. Codtoal relablty R ( x / T ) [ ( t x) ( t )] has bee coputed usg the equato R( x / t) e cosderg x0.. R( x / T ) ad FDR b at each cuulatve te are gve Table. The correspodg graph s gve Fg.. Table. FDR & Codtoal relablty correspods to cuulatve te Falure Te FDR (b) R(x/T) Falure Te FDR (b) R(x/T)

112 6 S. Chatteree L. N. Upadhyaya J. B. Sgh S. Nga: Cobed Effect Fgure. Fault Detecto Rate ad Codtoal Relablty w.r.t. Cuulatve te Cocluso I ths paper a attept has bee ade to study the cobed effect of creasg FDR ad decreasg FIR o error estato as well as relablty growth of software whe the debuggg process s perfect. It has bee observed that the assuptos ade about the FDR ad FIR are logcal. The odel ca be used other software falure data by assug proper FDR & FIR. The proposed odel ca be used for estatg release te ad cost of software. Acowledgeet Authors acowledges Uversty Grats Coucato (UGC) New Delh Ida for facal help the proect uber F.No.-5/007(SR). Also authors acowledge ISM Dhabad for provdg facltes to carry out the wor.

113 STATISTICS IN TRANSITION-ew seres Deceber REFERENCES []. MUSA J.D.; IANNINO A. ad OKUMOTO K.; Software Relablty Measureet Predcto Applcato McGraw-Hll It. Ed. 987 []. IE M.; Software Relablty Modellg World Scetfc Press 99. []. LU; M.R.; Hadboo of Software Relablty Egeerg McGraw-Hll: N 996. [] JELINSKI Z. ad MORANDA P.B.; Software Relablty Research Statstcal Coputer Perforace Evaluato W. Freberger. Ed. Acadec N.. 97 p [5] SHOOMAN M.L.; Probablstc Models for Software Relablty Predcto Statstcal Coputer Perforace Evaluato W. Freberger Ed. Acadec N.. 97 p [6] WAGNOR W.L.; The Fal Report o Software Relablty Measureet Study Report TOR-007 ()- The Aerospace Corporato El Segudo C.A. 97. [7] SCHICK G.J. ad WOLVERTON R.W.; A Aalyss of Copetg Software Relablty Model IEEE Tras. O Software Eg. vol. SE- 978 p [8] MUSA J.D.; A Theory of Software Relablty ad Its Applcato IEEE Tras. O Software Eg. vol. SE- 975 p. 7. [9] LITTLEWOOD B. ad VERRALL J.L.; A Bayesa Relablty Growth Model for Coputer Software Appl. Statst. vol. 97 p. 6. [0] SINGPURWALLA N.D. ad SOER. R.; Assessg (Software) Relablty Growth Usg A Rado Co-effcet Autoregressve Process ad Its Rafcatos IEEE Tras. O Software Eg. vol. SE- 985 p [] IE M.; A Shoc Model for Software Relablty Mcroelectro. Rel. vol p [] GOEL A.L. ad OKUMOTO K.; A Te-Depedet Error Detecto Rate Model for Software Relablty ad Other Perforace Measure IEEE Tras. O Rel. vol. R p.06. [] CHATTERJEE S; MISRA R.B. ad ALAM S.S.; Jot Effect of Test Effort ad Learg Factor o Software Relablty ad Optal Release Polcy ; Iteratoal Joural of Syste Scece; Vol. 8; No. ; 997; p

114 6 S. Chatteree L. N. Upadhyaya J. B. Sgh S. Nga: Cobed Effect [] SUMITA U. ad SHANTIKUMAR J.G.; A Software Relablty Model Wth Multple-Error Itroducto & Reoval IEEE Tras. O Rel. vol. R p [5] FAKHRE - ZAKERI I. ad SLUD E.; Mxture Models for Relablty of Software Wth Iperfect Debuggg: Idetfably of Paraeters IEEE Tras. O Rel. vol. 995 p. 0. [6] ZEEPHONGSEKUL P.; IA G. ad KUMAR S.; Software-Relablty Growth Model: Prary Falures Geerate Secodary-Faults Uder Iperfect Debuggg IEEE Tras. O Rel. vol. } 99 p. 08. [7] KAREER N.; KAPUR P.K. ad GROVER P.S.; A S-Shaped Software Relablty Growth Model Wth Two Types of Error Mcroelectro. Rel. vol. 985 p [8] IA G.; ZEEPHONGSEKUL. P. ad KUMAR S.; Optal Software Release Polcy Wth a Learg Factor for Iperfect Debuggg Mcroelectro. Rel. vol. 99 p [9] PHAM H.; A Software Cost Model Wth Iperfect Debuggg Rado Lfe Cycle ad Pealty Cost It. J. of Sys. Sc. vol p [0] KAPUR P.K.; SHARMA K.D. ad GARG. R.B.; Traset Solutos of a Software Relablty Model Wth Iperfect Debuggg ad Error Geerato Mcroelectro. Rel. vol. 99 pp [] CHATTERJEE S; MISRA R.B. ad ALAM S.S.; A Geeralzed Shoc Model for Software Relablty ; Coputer ad Electrcal Egeerg-A Iteratoal Joural Vol. ; 998 p.o.: [] ZHANG ; TENG ; ad PHAM H.; Cosderg Fault Reoval Effcecy Software Relablty Assesset IEEE Tras. O Systes Ma ad Cyberetcs-Part A: Systes ad Huas Vol. No. Ja. 00 p. 0. [] MISRA P.N.; Software Relablty Aalyss IBM Syst. J. Vol. 98 p. 6 7.

115 STATISTICS IN TRANSITION-ew seres Deceber STATISTICS IN TRANSITION-ew seres Deceber 009 Vol. 0 No. pp MONITORING WORKERS REMITTANCES AND BENEFITS IN UGANDA: THE STATISTICAL ISSUES E.S.K. Muwaga-Zae (PhD) ABSTRACT Due to the creasg portace of rettaces to Ugada efforts are uderway by the Cetral Ba ad Cetral Statstcs Offce the coutry to prove the regulatory ad otorg of evroet. A ultfaceted approach s used. Ths cludes the eactet of a ew law ad regulatos provg adstratve reportg ad carryg out surveys the aor rettg coutres ad Ugada. However these have ssues of collectg coplete accurate ad tely data.. Bacgroud Worers rettaces are curret trasfers by grats eployed aother coutry ad have lved those coutres for at least oe year. The players volved are aly related persos (IMF 99). I ay of the least developg coutres (LDCs) teratoal rettaces ow costtute the secod largest captal flow after Foreg Drect Ivestet (FDI). Rettaces costtute the fastest growg ad ost stable captal flow to developg coutres. The exact aouts of these flows are however ucerta ad the statstcal coplato of rettaces eeds proveet especally Sub-Sahara Afrca. Offcal rettace statstcs reported the balace of payets (BOP) typcally uderestate actual levels ad several coutres urecorded rettaces are sgfcat. (Terry F.D. et. al. 005). Isttute of Statstcs ad Appled Ecoocs (ISAE) Maerere Uversty P.O. Box 706 KAMPALA Ugada: (Forerly Drector Trade ad Exteral Debt Departet (TEDD) Ba of Ugada P.O. Box 70 KAMPALA) E-Mal: [email protected]; [email protected]. The vews expressed are those of the author ad oe of the two sttutos.

116 66 E. S. K. Muwaga-Zae: Motorg Worers. Why s the Ba of Ugada terested worers rettaces? I Ugada Balace of Payets (BOP) estates of rettaces for the perod 996 to 000 averaged % of exports. The fgure creased to 65% of exports durg the perod 00 to 005 whle for 006 the fgure s estated at US$ 65llo equvalet to about 6.5 percet of GDP (US$ 0 bllo). Worers rettaces are therefore the secod largest cotrbutor to foreg exchage flows after exports of goods ad servces thereby cotrbutg sgfcatly to the BOP exchage rate stablty ad also affectg oetary polces (Tuuse Muteble 006). More specfcally worers rettaces are portat because: a) They are a regular source of coe of recpet households ad spedg power for ther fales. b) They pact o poverty reducto ad welfare proveet through ther acro-ecooc effects. c) They crease vestets of households o educato captal for busess vetures ad health. d) They are less volatle sources of foreg exchage as they ted to be couter cyclcal creases tes of ecooc depresso poltcal turol ad atural dsasters ule other fors of flows le Foreg Drect vestet (FDI) ad Exteral Debt. e) They are a steady strea of Foreg exchage eargs that ca prove the coutres credtworthess. The Ugada Goveret through the Ba of Ugada (the Cetral Ba) coucto wth the Ugada Bureau of Statstcs (the Cetral Statstcs Offce) has ebared o a prograe to otor worers rettaces to ad fro the coutry. The aor characterstcs of rettaces eed to be clearly establshed order to prove the uderstadg of rettace pacts. Studes ad surveys are requred both the sedg ad recevg coutres.. Coceptual ad Measureet Issues of Rettaces The ecooc sgfcace of rettaces s ot fully captured the offcal BOP statstcs ether sedg or recevg coutres. Moetary trasfers crease the supply of foreg exchage the coutry of org of the grat whle -d rettaces the for of goods ad servces save scarce foreg exchage the recpet coutry. The easureet of both cash ad -d rettaces has prove to be very dffcult precse ad coplete. Oly soe of these trasactos are recorded. Where rettaces are set through foral chaels they are recorded by the recevg coutry s offcal statstcs as a flow the curret accout of the BOP. Coversely cash rettaces set forally through courers are usually

117 STATISTICS IN TRANSITION-ew seres Deceber urecorded offcal statstcs. I-d rettaces or goods ad servces set to households the hoe coutry or brought o ther retur hoe ay be oly partally captured as ports offcal data. Very lttle data exsts o the sze of rettaces d. Other trasfers the for of chartable doatos or payets ad deposts for relatves ad freds (such as surace preus tuto ad travel costs) fucto as a ecooc for of rettaces but are rarely recorded as such. Choosg betwee offcal or uoffcal trasfer chaels s a portat decso whe sedg rettaces ad depeds o several factors. These clude the avalablty ad adequacy of bag servces trasacto costs ow-yourcustoer requreets ad the potetal for addtoal eargs through uoffcal chaels. Whe offcal eas of rettace trasfer are uattractve the prvate sector or groups of egrats theselves ofte set up parallel systes. Dealers foral trasfer systes soetes provde copettve exchage rates as well as other assstace ad servces of a persoal ature. All these ea that the data Ugada s coplete ad accurate. It s ot possble to accurately dsaggregate offcal estates of rettaces by source due to the resdual ethod of estato whch s used ag the estates.. What s Beg Doe Ugada? The Ba of Ugada (BOU) s usg a ultfaceted approach of easures outled below aed at provg the recordg aageet ad tegrty of worers rettaces whle at the sae te tryg to axze the beefts ters cotrbutg to ecooc developet... Publcty ad awareess capags A sestzato capag aogst Ugadas the Daspora ad the recpets ad servce provders the prvate sector s ogog. The capag ephaszes trasparecy copetto ad custoer forato ad s desged to hghlght polcy ad operatoal ssues. The capag also wors through the Ugada ebasses ad ethc group based assocatos. The Ba s also cosderg creatg ls wth web stes of the two Ugada ewspapers whch are wdely read by the Ugadas the Daspora... Legal ad Isttutoal Fraewor for the Moey Trasfer Busess The Ugada Goveret has revsed the law o foreg exchage trasactos to provde for ore oey rettace servces thus ag t easer for Ugadas abroad to sed oey hoe (Ugada Goveret 005). Apart fro coercal bas ad other facal sttutos Rettace Lceses ay

118 68 E. S. K. Muwaga-Zae: Motorg Worers ow uder certa codtos be obtaed by Iteratoal Moey Trasfer Ageces ad Foreg Exchage Bureau. BOU has also facltated otable proveets the payet syste to ae t ore effcet ad wde-spread. These efforts clude: stallato of electroc bag wth the atoal electroc swtch; 0 Autoatc Teller Maches (ATMs) spread coutrywde; ad the troducto of credt cards ad debt cards. Soe facal sttutos are already cosderg troducg Iteret bag. Moble Bag Servces are expected to reach a estated 0000 clets over a perod of fve years. All these developets should facltate a soother flow of rettaces ad attract rettaces to the foral syste ad thereby ove soe of the htherto foral rettaces to the foral (offcal) chaels ad hece ae the easer to record. Ths wll prove data collecto. To cap all these BOU created a Trade Statstcs ad Rettaces Dvso the the Trade ad Exteral Debt Departet (TEDD) to cocetrate all efforts the estato of rettaces... New Methods of Reportg/ Reportg Requreets for Moey Rettace Busess The ew law stregthes the capacty of BOU to otor ad regulate the trasactos. Uder ths law all lcesed retters are expected to ae weely ad othly returs to BOU ad therefore provde forato o rettaces o a cotuous bass. Ufortuately detals for rettaces through coercal bas ad other facal sttutos are dffcult to detfy. Slarly returs fro the ewly-lcesed retters are stll faulty ad coplete. As at 0th Jue 007 there were twety eght (8) No Bag Facal Isttutos (NBFIs) lcesed to egage oey rettace busess coprsg: forex bureaus four () Mcro Face Depost-Tag Isttutos ad two () Credt Isttutos. Facal Isttutos (Bas) lcesed uder the Facal Isttutos Act 00 are exepted fro coplyg wth the provsos of the Foreg Exchage Act 00. All Moey Retters (MR) lcesed uder Secto 5 of the Foreg Exchage Act 00 are requred to subt to Ba of Ugada the followg returs: Weely (Sed ad Receve) Mothly (Sed ad Receve) ad a suary Mothly trasactos as specfed Secto 7 () ad Schedule 8 of the Foreg exchage (Forex Bureau ad Moey Rettace) Regulatos 006. All the 8 NBFIs are subttg these returs as specfed by the regulatos. Coplato of the returs to eagful reports coeced the secod quarter of 007 ad as at 0th Jue 007 total flows aouted to US$7.0Mllo whle outflows captured were US$0.585Mllo. Ths s a clear dcato that ether the foral eas of rettg oey s ot use as expected or the ode of capturg the data s ot deal/ accurate. It ay therefore be preferable to capture such forato fro the source coutres ad through

119 STATISTICS IN TRANSITION-ew seres Deceber coercal bas sce the evetual settleet of such cross-border trasactos s through a coercal ba... Surveys o Rettaces Although balace of payet data o rettaces are cooly used for estatg the volue of rettaces they could be sleadg sce foral rettace flows are ot accouted for these calculatos. The Ba of Ugada collaborato wth the Ugada Bureau of Statstcs (UBOS) have sttutoalzed surveys o rettaces to Ugada. The obectve s to estate the specfcs of the rettaces ters of source ad sze of rettaces frequecy ad chaels of rettg use of rettaces seasoal patter the deographc characterstcs of the recpets ad the locato of the retters the soco-ecooc codtos ad tetos of the seders etc. Surveys have bee carred out the aor sedg coutres begg wth the UK the USA ad South Afrca. The rettaces to Ugada are aly fro the UK the USA Japa South Afrca ad Swede. Iforal servce provders reveal that the UK the USA ad the Scadava coutres accout for about 60% 0% ad 0% of trasfers respectvely. I order to ehace the exstg owledge o rettaces to developg coutres the Facal Maret Itegrty Ut (FPDFI) has lauched a seres of studes called Blateral Rettace Corrdor Aalyss (BRCA). The teto s to use the owledge gaed through these studes to develop best practces to protect the tegrty of rettace arets ad to prove effcecy ad trasparecy of trasfer chaels for rettace flows. Ths study was tated at the request of the Ba of Ugada (BoU) as a World Ba-BoU ot study. The purpose of the ot study s to share owledge ad expertse of World Ba s Blateral Rettace Corrdor Aalyss (BRCA) wth the Ba of Ugada. For ths purpose a sso was otly coducted the Uted Kgdo ad the Uted States. For these ssos the Gera Gesellschaft für Techsche Zusaearbet (GTZ) provded facal support. Beg the frst BRCA study to be coducted wth the partershp of a local authorty adds to the sgfcace of ths study. The study evolved fro a orgal Uted Kgdo Ugada rettace corrdor to three corrdors cludg the US-Ugada ad South Afrca-Ugada. These two corrdors were cluded cosequece of a suppleetary request ade by the Ba of Ugada. Ital rettace data fro a plot survey dcated that a large volue of rettaces sees to orgate the Uted States ad aecdotal evdece of rettace flows fro South Afrca. The South Afrca- Ugada corrdor added value to the study sce t allows the study to copare North-South corrdor (UK-Ugada ad US-Ugada) ad South-South corrdor. The a obectve was to descrbe how characterstcs of the three rettace corrdors affect rettace flows to Ugada ad plcatos of such characterstcs

120 70 E. S. K. Muwaga-Zae: Motorg Worers o the desred shft fro foral to foral rettace flows. I order to acheve ths obectve aalyss was ade for the rettace seders rettace flows ad aret players regulatos rss of oey lauderg ad the exstg At-Moey Lauderg ad Cobatg the Facg of Terrors (AML/CFT) fraewor ad challeges to expadg access to facal servces. Based o the aalyss the report provdes a set of focused polcy recoedatos to regulators ad aret players order to foster a copettve ad trasparet foral aret. Ths report s the culato of the efforts ad cotrbutos of FPDFI Cetral Ba of Ugada ad The Gera Gesellschaft für Techsche Zusaearbet ad ay others. The feldwor was coducted Ugada the Uted Kgdo the Uted States ad South Afrca. They cluded tervews wth the authortes teratoal orgazatos prvate sector ettes (bas oey trasfer operators ad research orgazatos) No-goveretal orgazatos ad Ugada grats..5. The Ugada Natoal Household Survey (UNHS) 005/06 Durg the UNHS 005/06 carred out by UBOS forato was collected o recept ad use of both doestc ad teratoal rettaces at household level. As show Table overall proporto of recpets of rettaces fro local sources (%) was uch hgher tha that for rettaces fro abroad (%). The captal Cty Kapala had the hghest proporto of households recevg rettaces fro abroad (7%). The atoal ea othly value of rettaces fro abroad was about US$0. Table. Households that receved a Rettace durg the last oths by Resdece (%) Resdece Percetage of households fro doestc sources fro abroad Mea othly Value* of aout receved (USh) fro doestc sources fro abroad Rural/Urba Urba Rural Rego Kapala Cetral Easter Norther Wester Ugada *Note: The Value of rettaces cludes both cash ad - d. Source: UNHS 005/06

121 STATISTICS IN TRANSITION-ew seres Deceber As Table shows ost recpets used the rettaces to purchase cosupto goods ad servces rrespectve of ther source. Ths was followed by payet for educato expeses. It reforces the fdgs by Ba of Ugada that the shllg teds to apprecate durg the te of payg school fees ad Chrstas holdays whe the grat worers sed fuds to support ther fales. Table. Recpets by Purpose ad Source of Rettaces (%) Ma Purpose of Rettaces Source of Rettaces doestc abroad Purchase cosupto goods ad servces Pay for educato expeses.6 6. Pay for health expeses Worg captal for o-far eterprses Purchase buldg aterals Buy lad 0..5 Buy far puts tools ad pleets Pay for cereoal expeses..0 Other. 5.9 Total Source: UNHS 005/06.6. Plot Survey 006 A plot survey was carred out Jauary 007 to test the questoare for the subsequet coutrywde survey o rettaces. The prelary fdgs revealed that alost 50% of the beefcares receved uder US$00 per au. The sae percetage of the respodets dcated preferece to use foral chaels for sedg oey to Ugada. The orgs of rettaces are dverse but the UK ad the USA doate. The results also showed that rettaces are used aly for cosupto health care chldcare ad educato. There s also a otceable use of rettace fuds for busess support ad real estate. For exaple 6% of the recpets dcated that rettaces are spet o buldg wors ad lad purchase..7. Survey o Rettaces to Ugada 007 The a obectve of the atowde survey otly coducted by BOU ad UBOS was to collect hgh qualty data o rettaces receved 006 at household level so as to prove the accuracy of the BOP ad therefore stregthe the forulato of oetary ad exchage rate polces. There was

122 7 E. S. K. Muwaga-Zae: Motorg Worers also a couty questoare to detere the cotrbuto of rettaces at couty level. The survey covered over 000 radoly selected households. The survey sought to establsh a accurate estate of the value ad volue of rettaces receved durg 006 the org of such rettaces ad the characterstcs e.g. aouts frequecy use of receved rettaces receved. The results are outled below. The up-rated estated rettace value fro the survey of US$06.5 llo was below the offcal estate (US$665) for the year 006. However the offcal estate was derved as a resdual fro BOP coputatos. Ths was the coutry s frst attept eprcal assesset of the sze of ward rettaces hece ths estate was thought to be ecouragg ad cosdered to be dcatve of the agtude of such flows. There were also detfable ethodologcal gaps whch could be addressed through refeet of saplg procedure ltg the geographcal scope of the survey revso of the survey struets ehaced trag of feld staff a strategc sestzato prograe to create awareess ad buld cofdece the respodets aog others. More tha half of the recpets receved rettaces oce or twce a year. Ths fdg s supported by the fact that Deceber ad Jauary retured the hghest proporto of rettaces durg 006 cofrg the practce of grats sedg oey hoe durg the festve seaso ad school related expedture. However wth oe roud of feldwor t ay ot be possble to coclude o the regularty of wards rettaces. I addto to oetary rettaces Ugadas also receved rettaces d ostly oce a year ad these were aly clothes ad doestc applaces. Such goods suppleet household coes. However estato of values for d rettaces was rather coplcated due to osso of values at the oset. Future surveys should provde for ths d of estato. The sources of rettaces to Ugada were dverse wth Europe ad Afrca beg dstct ters of share of total volue. Ths fdg ay be explaed by the exstece of hstorcal tes addto to proxty ad ease coucato betwee Ugada ad these regos. The results also revealed presece of southto-south rettaces to Ugada wth 7 percet of the respodets dcatg that they receved rettaces fro Afrca. The fdgs o educato ad age revealed that the aorty of retters are youg educated Ugadas. At the sae te ore tha half of the retters had lved abroad for betwee 5-0 years. These fdgs are a dcator of the cotued oveet of Ugadas search of eployet abroad. Over 65.5 percet of the respodets dcated that they would rather use foral chaels for recevg rettaces a good dcator of how the populato perceves ad apprecates the foral facal servces. The a reaso for ths preferece s safety of the rettaces. Ths clato towards foral facal servces should be leveraged the facal deepeg strategy ad furtherace of the payets systes Ugada. However ay Ugadas stll use foral

123 STATISTICS IN TRANSITION-ew seres Deceber ethods that s freds ad acquataces for rettace trassso whch ay be attrbuted to avodace of trasacto costs assocated wth foral chaels. There s therefore eed to sestse users o the varous servce provders avalable the type of servces that they are authorzed to egage ad the rss assocated wth the use of foral ethods of rettg. Rettaces beeft the retters ad recpets. However there s a dstct dfferece the respectve use wth cosupto for recpet ad vestet for retter. Whle Retters ret to ata the edate ad exteded fales they persoally see to have a terest log-ter developet proects. Though the survey dd ot drectly as a questo o beefcary sectors fro the expedture patter oe ca easly l expedture wth three aor sectors aely wholesale ad retal trade-for cosupto goods educato ad health for the beeft of both recpets ad retters. I addto to these three vestet stads out cases where rettaces beeft the retter. The report detfed the uder-lsted lessos leart ad areas of focus for future surveys; The eed to revst the ethodology to facltate better estato of the agtude of rettaces: The saple frae: Rettaces see to orgate urba areas ad dstrbuto taes place thereafter. It would therefore be of beeft to coduct a urba survey o rettaces; Assesset of regoal dstrbuto ad pocets of rettaces f ay; ad Assesset of the three-ter dstrbuto syste ters of role of servce provders alog the cha ad aret share of each. I order to buld o the survey results there s eed to coduct slar studes o aual bass. Ths ay be followed by cesuses after every 5 years. Subsequet surveys are also ecessary to cofr the revealed seasoal patter of rettace flows. Assesset of the pact of rettaces calls for drawg coparsos betwee households that receve ad those that do ot receve rettaces. Ths ay be acheved through the cesus. A depedet survey o outward rettaces wth approprate ethodology s ecessary to coplete the coutry s posto o worers rettaces for Balace of Payets. Icorporate cotuous sestzato the rettace otorg strategy ot oly for the survey actvtes. Rettaces d are a reasoable proporto of total flows. It s portat to address valuato ssues for better estato of the value of rettaces d.

124 7 E. S. K. Muwaga-Zae: Motorg Worers.8. Fdgs the Uted States of Aerca (USA) There s a aual eetg of the Ugada Norther Aerca Assocato (UNAA) durg the Labour Day Wee-ed. These eetgs are atteded by over 000 Ugadas leavg the USA Caada the Carbbea the UK ad soetes South Aerca to share deas challeges ad also fellowshp. BOU has partcpated these eetgs sce 00 to dscuss rettaces. A survey was carred out durg the coveto New or 006 as part of the ogog ot World Ba/Ba of Ugada study o the USA-Ugada rettace corrdor. A total of questoares were retured. More tha half of the retters were graduates wth a frst degree the aorty havg lved the USA for perods ragg fro 0 to 0 years. Wester Uo was the ost preferred rettace chael wth 9% followed by MoeyGra (8%) ad the bas (7%). Maor reasos or cosderatos for choce of edu ( order of portace) cluded cost/rates coveece speed ad safety/relablty of the gve edu. Oly 6% use foral chaels. Costs are wth -% of the total retted; rettace through bas was oted to be ore costly as t volves payet of fees at both the sedg ad recevg pots. Retters prefer to use the oey trasfer copaes for lower ad bas for the bgger aouts. Faly support was the aor use (teded) of rettace as dcated by 59% of retters whle alost 7% were equally dvded betwee educato ad vestet/busess proects. The ea rettace value was about US$700. The 9th UNAA coveto was held fro 0th August to rd Septeber 007 Sa Fracsco Calfora. Ths ad future UNAA covetos should provde a platfor for feedbac o the Ba s fdgs ad ore elaborate surveys o rettaces..9. Fdgs the Uted Kgdo Rettaces servce provders the UK clude bas oey trasfer copaes ad foral provders. The Facal Servces Authorty regulates the bas whle the oey trasfer copaes are regstered regulated ad supervsed by Her Maesty s Reveue ad Custos (HMRC). There are ethc group based oey trasfer servce provders to Ugada regstered wth HMRC uder the Moey Lauderg Regulatos (00). These retters have fored a assocato whch could be used to collect data o the aouts of rettaces. However a few uregstered dvduals are ow to offer rettace servces. The dvduals operate outsde the defed regulatory rege characterzed by easy etry ad operatoal requreets. A prelary study has bee carred out o the regstered retters the UK. The study gave soe data o the volues retted ad the costs of rettg. It cofred that soe of the rettaces were very low. Rettace values of betwee 0 ad 6 pouds (US$9-0) are ot ucoo for rettaces

125 STATISTICS IN TRANSITION-ew seres Deceber chaelled through ethc group based servce provders. They are preferred because of the relatvely low rates charged addtoal persoal servce offered less paper wor ad hgher speed of delverg servces. Trasacto charges rage betwee 7 to 5 percet of the rettace value through bas ad teratoal Moey Trasfer Orgazatos whle ethc group based oes charge percet of the value..0. Fdgs fro South Afrca I South Afrca MoeyGra s the oly teratoal MTO provdg rettace servces by collaboratg wth bas. Quatfyg the rettace flows s a challegg tas resultg fro lac of proper ethodologes accuracy of recorded data ad extesve use of foral echass. South Afrca seds ad receves cross-border rettaces wth ultple rettace corrdors wth ad outsde Afrca. As a sedg coutry South Afrca s the largest source of rettaces Sub-Sahara Afrca. The volue of rettace flows fro South Afrca to Ugada s uow. Ule the forato of Ugadas the Uted States ad the Uted Kgdo there s lted data avalable o the Ugada populato South Afrca. The lac of forato wth regard to the sze of per capta rettace the sze of populato ad coe level aes t dffcult to develop a estate of the rettace volues... Suary of Fdgs the Three Blateral Rettace Corrdors Ugada s ot a aor rettace destato the three coutres although these rettaces are portat to the Ugada ecooy. Rettace data are ot avalable fro South Afrca whch s doubly portat as a South-South corrdor as well as a regoal rettace corrdor. Dfferet grato geeratos of both docueted ad udocueted Ugadas have etered the three corrdors over the past 5 decades. Idetfyg the sze of the udocueted Ugada grat populato s dffcult ag estatg rettaces flows coplcated. Igrato polces of the three coutres have ade t dffcult for low-slled grats to see teporary eployet these coutres; hece ay low-slled worers do ot have proper grato docuetato whch leads to lted or o access to foral facal servces. Access to rettace servces vares although docueted grats have full access to foral facal servces whle udocueted worers all three coutres have lted or o access to foral facal servces. The udocueted Ugadas the Uted Kgdo have

126 76 E. S. K. Muwaga-Zae: Motorg Worers access to foral rettace facltes f they have ay goveret-ssued ID cludg Ugada atoal detfcato. Ivestg busesses ad preparg for retreet Ugada addto to supportg ther fales s hgh o the ageda of the Ugada Dasporas all three coutres. Hece ost Ugadas show prefereces to eep closer tes wth ther hoe coutry. Regstered oey servce busesses are allowed to provde rettace servces the Uted Kgdo ad the Uted States; o the other had South Afrca oly authorzed dealers aely bas ad foreg exchage bureaus (ost of these ettes are part of bas) are allowed to coduct rettaces. I South Afrca oly MoeyGra aog teratoal MTOs provdes rettace servces by collaboratg wth bas. Rettace costs the Uted Kgdo ad South Afrca are substatally hgher tha the costs the Uted States. The foral rettace aret South Afrca lacs copetto wth oly oe teratoal MTO coercal bas ad ther subsdary foreg exchage bureaus. I the UK Ugada corrdor teratoal MTOs have a advatage overall prcg whle ethc MTOs are ore copettve. I the US Ugada aret teratoal MTOs provde copettve servces sce they cosder Afrca as a sgle aret ad are copettve all corrdors to Afrca. AML/CFT laws ad regulatos exst all three coutres however they are ot slar. I the area of rettaces the FATF requres coutres to esure that oey trasfer busesses are regstered or lcesed. The Uted Kgdo ad the Uted States requre oey servce busesses to be regstered; South Afrca requres the to be lcesed. The Uted Kgdo ad South Afrca have a cetralzed regulatory fraewor for the rettace aret whle US regulatos are frageted aog federal ad state levels... The Afrca Developet Ba Worshop May 009 It was oted that data o rettaces was tally collected as a resdual of captal flows after subtractg explaable flows such as exports ad curret trasfers. More drect ethods of estatg rettaces have sce bee adopted partcularly the use of surveys sttutos through whch rettaces flow.e. bag sttutos ad copaes regstered to do the busess of oey trasfer. It s beleved that the ew approaches produce ore accurate data o rettaces. The challege Ugada s facg s that such surveys have bee peceeal. The ey recoedato was to prove data collecto o grato ad rettaces through a coprehesve survey to get a correct pcture of rettaces Ugada.

127 STATISTICS IN TRANSITION-ew seres Deceber Cocluso: Challeges ad the Way Forward Fro the above fdgs o rettaces to Ugada the followg ssues o the estato of the rettaces to Ugada ca be detfed: () The preferece for use of foral chaels for varous reasos ad the attedat dffcultes obtag data strogly suggest that avalable data o rettaces could be uderstated whch calls for proveet ethodology estatg the volue of rettaces. Iforalty s explaed by access ssues shallow bag syste; poor techology; ad low labour oblty expla the persstece of the foral syste rettaces. () Issues of cost less cubersoe paper wor ad persoalzed servce see to attract users to the foral syste. The foral chaels eed to focus o the provg the delvery of these preferred characterstcs of grat rettaces. Whle the Kow-our-Custoer requreets are ecessary for protectg the tegrty of the facal sector these clearly coe out as costrats the access to ad use of foral chaels. Harozato of the ore restrctve supervso ad regulato of the facal sector at the recevg ed wth the ore accoodatve rege at the sedg pots s crtcal for creasg the use of foral chaels of oey trasfer. () The use of cash as the aor delvery struet hghlghts the effceces ad the eed for further developet of the payets syste. (v) The aual UNAA Meetgs USA ad the Retters Assocato the UK preset uque opportutes to collect data o rettaces to Ugada fro the two aor source coutres. Obvously t wll be ecessary to cover retters who do ot atted the UNAA coveto or who do ot ret through the Retters Assocato. Ugadas Japa have also fored a Assocato ad are plag a UNAA-le covetos. (v) More wor eeds to be doe to get coplete ad tely data fro the foral retters cludg coercal bas ad other facal sttutos. It s also ecessary to l rettaces to poverty reducto ad establsh the seasoal patter for purposes of exchage rate aageet ad coduct of oetary polcy. (v) The dverse org sall values ad use to whch rettaces are chaelled presets the challege of detfyg the respodets ay survey. However the cocetrato of recevg pots the captal cty Kapala ay ea that the surveys could be cocetrated there to beg wth. (v) The collecto of data o rettaces s a further llustrato of the collaborato efforts the Ugada statstcal syste. Other exaples clude the collecto of data o Prvate Captal Flows (PCF) Iforal Cross Border Trade (ICBT) etc. Ugada has draw up a Pla for Natoal Statstcal Developet (PNSD).

128 78 E. S. K. Muwaga-Zae: Motorg Worers REFERENCES ABUKA CHARLES (008): The Goveret ad Rettaces: Ugada s Polcy. Presetato at the Worshop o Rettaces Ugada: Afrca Developet Ba Group; 8 May 008. Afrca Developet Ba (ADB) Rettaces Ugada: The Way Forward Serea Hotel Kapala Ugada 8 May 009; WORKSHOP REPORT. Ba of Ugada Ugada Bureau of Statstcs & GTZ; (007); Iward Rettaces 006; Survey o Rettaces to Ugada 007 Worers Rettaces Report. Iteratoal Moetary Fud (99); Balace of Payets Maual; Washgto D.C. MUWANGA-ZAKE E.S.K (00): Trasferrg Fuds to Ugada: The Curret Legal Ways ad Issues Paper Preseted at the UNAA Coveto Mesota Septeber 00. OROZCO MANUEL (008): Rettace Trasfers ts aretplace ad Facal Iteredato Ugada: Prelary Fdgs Lessos ad Recoedatos. Report Cossoed by the Iter-Aerca Developet Ba co-operato wth the Afrca Developet Ba. TERR F. DONALD; WILSON R. STEVEN (Edtors) (005); Beyod Sall Chage: Mag Mgrat Rettaces Cout; Iter-Aerca Developet Ba Washgto D.C. TUMUSIIME MUTEBILE EMMANUEL (006) Goveror Ba of Ugada; Speech at the Secod Iteratoal Coferece o Mgrat Rettaces: Rettace ad Access to Face; Couty Hall; Lodo Uted Kgdo; Noveber 006. Ugada Goveret (005); The Foreg Exchage Act 00 ad the Foreg Exchage (Forex Bureaux ad Moey Rettace) Regulatos 006. Ugada Bureau of Statstcs (006); Ugada Natoal Household Survey 005/06. World Ba (008) Blateral Rettace Corrdor Aalyss (BRCA) Rettace Corrdors to Ugada Uted Kgdo Uted States ad South Afrca: Challeges to lg rettaces ad use of foral facal servces. Isau Edo ad Jae Naaa; Edtors.

129 STATISTICS IN TRANSITION-ew seres Deceber STATISTICS IN TRANSITION-ew seres Deceber 009 Vol. 0 No. pp ANALSIS OF TEMPORAR ASPECTS OF POVERT IN POLAND BETWEEN B HAZARD MODELS Natala Nehrebeca ABSTRACT Ths paper exaes the extet ad characterstcs of poverty Polad aalyzed o the bass of pael data fro CHER (Cosortu of Household Paels for Europea Soco-Ecooc Research) for the years The aalyss preseted shows a low households dyac of coe ths perod. The total uber of years spet poverty as well as dfferet sequeces of etry to ad ext fro poverty suggest the tedecy to a persstet for of ths pheoeo the populato. Durg the perod studed the bass for the calculato of the uber of years spet poverty was the rate of ext fro ad etry to poverty. The calculatos have bee ade accordg to the ethod of aalyzg poverty dyacs by hazard odels cosderg observed ad uobserved heterogeety of dvduals to expla a chace of ext ad retur to the sphere of poverty. Key words: Peraet ad traset poverty Hazard odels Mult-spell poverty epsodes Uobserved heterogeety Itroducto Poverty has affected socety sce the begg of ad. It stll s expereced ot oly by ctzes of udeveloped coutres but also by those lvg hghly developed ecooes. The exstece of poverty o a large scale leads to a wde terest ths topc although research o poverty s prarly coducted by coutres whch the scope of ths pheoeo s argal. A dfferet stuato perssted cetrally-plaed ecooes where poverty was ot offcally recogzed as the a of socalst reges was to elate poverty. I recet years coe dyacs ad durato of poverty have bee ore ofte dscussed durg publc ad acadec dscussos as socally portat Departet of Statstcs Natoal Ba of Polad Faculty of Ecooc Sceces Uversty of Warsaw e-al: [email protected] [email protected]. I would le to say specal tha to prof. B. Górec prof. M. Wśews for excellet suggestos fro whch I beefted a lot.

130 80 N. Nehrebeca: Aalyss of teporary factors copletg forato o coe dstrbuto. It has bee agreed that the best way to easure ad uderstad the proble s to suppleet tradtoal studes based o cross-seres data wth log-ter aalyss based o pael data. Whe a household s gross coe decles below a u level ad there are ot eough facal resources to satsfy basc hua eeds ts ebers are cosdered to be poor. Whe ths stuato perssts for a exteded perod of ther lves the a etre socety s affected by effects of poverty. A log-ter poverty of the sae group of people s dffcult to accept ot oly because of socal faress but also due to hgh exteral costs of peraet argalzato ad socal excluso. Poltcal stablty s also a portat reaso why the topc of poverty arouses so uch terest both hghly developed ad developg coutres. The a of ths paper s to aalyze durato of poverty Polad over the perod of usg pael data fro the CHER database. A exact specfcato of the populato peraetly lvg poverty ad ts dfferetato fro the category of people who experece ths state teporarly ay ad preparg progras of socal assstace targeted at these groups that are ot able to ext poverty by theselves. Studes of ths type are ecessary due to a large scale of ths pheoeo as well as lted aout of resources whch for the sae of a peraet budget defct proble Polad ay be of use to overcoe the proble. If I succeed defg the category of those chrocally lvg poverty ths aalyss ay prove helpful targetg the facal assstace for the poor ore effectvely whle cuttg the budgetary resources earared for ths purpose at the sae te. The a questos put forth ths paper are as follows: Whe does a household fd tself poverty ad how log does the poverty last? Does recurrece of poverty exst.e. whether repeated spells of poverty could be cosdered a adequate dcator of peraet poverty? What soco-ecooc characterstcs of households allows for the best detfcato of poverty Polad betwee ? Ths artcle uses ecooetrc ethods of poverty aalyss by odel hazard tag to accout observable ad uobservable heterogeety of dvduals. It carres out estato of two copletary log-log type odels dscrete te. They dffer ethod by whch they descrbe durato of so-called base hazard fucto. I the frst odel t has a o-paraeter shape whle the secod t s a polyoal te fucto. The artcle cossts of fve parts. The frst part cludes the ethodology used costructg the ecooetrc odel. The secod part descrbes the data used the aalyss defto of poverty as well as depedet varables of the odel. The thrd part provdes a short descrpto of poverty dyacs ad ts durato Polad. To troduce a dstrbuto of te spet poverty a ext rate fro ad Cosortu of Household Paels for Europea Soco-Ecooc Research.

131 STATISTICS IN TRANSITION-ew seres Deceber a retur rate to poverty have bee used. The paraetrc ethod of aalyzg poverty s preseted the fourth part. The last part s a suary of the results ad coclusos.. Durato theory To study the ext fro ad retur to poverty rates of households ths artcle apples the Jes (995 00) ad Devcet (000 00) ethodology. The hazard fucto h(t) s used for calculatg ext ad retur probabltes. Let τ τ be a observato perod of the saple whle t t - perod of poverty. I the study oe ca have cesored observatos - that s those that cota forato how log the perso was affected by poverty but the exact oet of ext fro ad etry to poverty are uow. Let T be the legth of the poverty epsode of -the dvdual. Let the legth of ths te be a realzato of the cotuous rado varable T wth the dstrbuto fucto F(τ) ad the desty fucto f(τ) the the probablty that a state of poverty wll last shorter tha τ years uder a codto that t occurred s: F( τ) Pr( T < τ) However the probablty that occurrece of poverty wll last a u of τ years uder codto that t occurred s : S( τ) F( τ) Pr( T τ) where: S (τ) survval fucto. The cotuous te hazard rate s: θ(τ) Pr( τ T < τ Δτ T Δτ τ) l l Δτ 0 Δτ 0 () () f ( τ) () S( τ) Ths rate ca be terpreted as a probablty of edg the epsode of poverty the rage [τ τ Δ τ] for sall Δτ uder codto of survval to te τ. It s portat to otce that hazard rate cotuous te does ot satsfy all propertes of probablty ad especally the hazard rate ay be greater tha. The survval fucto as well as hazard rate are coected by sple drect relato that s: S ( θ) d θ( τ) l S( τ) () S( θ) dτ F( τ Δτ) F( τ) f ( τ) ΔS( τ) F( τ) h(t) hazard rate dscrete te θ(τ) hazard rate cotuous te. The survval fucto provdes a probablty that the perso wll lve loger tha soe gve te τ or other words that they lve utl te τ (Stasz 00).

132 8 N. Nehrebeca: Aalyss of teporary ad Ths research had begu fro o-paraetrc estato of survval ad hazard fuctos utlzg the Kapla-Meer estator... Kapla-Meer Estator τ S( τ) exp θ(z)dz 0 The Kapla-Meer estator s calculated accordace wth the followg forula : Sˆ( τ ) τ < τ h where: { τ :... } - s a set of all oets of the evets whch occurred; τ < τ <... < τ < - the order of durato of the epsodes; a uber of observatos < because a part of the observatos s cesored; h a uber of copleted cases wth a durato of τ ; a cout of the rs set that s a uber of epsodes exposed to copleto a te τ ( h ) ( h )... ( h ) ; where: - a uber of cesored observatos wth a cesored durato the terval [ τ τ ) ; λ ( τ ) a probablty of edg the occurrece the ear (rght-sded) τ pot uder a codto that the epsode has lasted utl pot τ. λ( τ ) s a ( ˆ( λ τ )) τ < τ (5) (6) A o-paraetrc aalyss allows us to aalyze data wthout ag assuptos regardg dstrbuto. Ths has certa beefts as well as ptfalls. O oe had a possblty to aalyze data wthout ag assuptos about the real dstrbuto of lfe lets oe overcoe potetally great errors. O the other had cofdece levels related to o-paraetrc aalyss are geerally uch wder tha those based o paraetrc calculatos; oreover the latter case forecasts beyod the saple are possble Kefer (988).

133 STATISTICS IN TRANSITION-ew seres Deceber theoretcal hazard fucto ad h ˆλ ( τ ) s a eprcal estator of hazard fucto at pot τ. The Kapla-Meer estator ca also be set usg the lfe durato table ethodology. I ths case the estator has a for of: where: d a uber of occurreces whch has fshed rage; r r ; where: a uber of cesored occurreces a rage a a ) ; d Sˆ( ) r ( r a rs set - a set of uts that lved to the upperost part of the rage ( a a ) ad so have a chace to eter ( a a ) rage. Therefore oe ca see that here a survval fucto s estated ad ts oly arguet s te. Based o survval fucto oe ca easly trasfor to hazard fucto of the followg for: θ.. The paraetrc ethod: dscrete te odel cludg observed ad uobserved heterogeety of dvduals d dτ Whe estatg the hazard rate wth the use of the Kapla-Meer ethod oe does ot cosder the heterogeety of dvduals whch depeds o observed ad uobserved varables. Ths proble s resolved by the paraetrc ethod. Although ext fro poverty ca occur at ay oet of te (stochastc process for cotuous te) usually the durato of poverty epsodes s observed dscrete ot cotuous te. The odels wth a dscrete te however have soe advatages. Oe of these coes fro the fact that dscrete te odels () τ l Sˆ( τ) (7) (8) Ths ethod s the oldest techque of estatg survval ad hazard fuctos. Jes (00).

134 8 N. Nehrebeca: Aalyss of teporary cobe varablty te wth elastc specfcato of durato terdepedeces. For the eeds of ths aalyss dscrete odels preseted by Jes (995) have bee used: ) Pretce-Gloecler odel (978); ) a exteded odel by Meyer (990) the Pretce-Gloecler odel (978) whch cotas the gaa dstrbuto that cludes uobserved dvdual heterogeety. A ext hazard rate fro a gve state (poverty or wealth) a dscrete te for -th dvdual perod t s specfed by Pretce ad Gloecler (978) as: h ( t) exp[ exp( θ 0 ( t) β ' ( t))] where: (t) depedet varables (varable wth te or fxed); β a vector of uow paraeters; θ 0( t) a base hazard rate that s a hazard for a gve dvdual whe all depedet varables values are equal to zero. The odel also ow as the copleetary log-log ca be terpreted as a odel wth dscrete te whch s drectly related to hazard cotuous te. The assupto of a base hazard for θ 0( t) ay uecessarly lt the scope of hazard ad brg potetal bas of the β estator. That s why t s especally portat to clude geeral oparaetrc specfcatos.. Data ad varable descrpto ad defto of poverty arg The data used ths eprcal research coe fro the CHER database. It s a harozed ad stadardzed croecooc database created fro already exstg paels pertag to lvg codtos of dvduals ad households the Europea Uo before ts expaso 00 as well as for Polad ad Hugary. The database cotas detaled data o coe ad professoal actvtes of dvduals ther educato eployet eployet hstory ad others. Varables descrbg socal relatos ad the setets of the ebers of the households are also cluded there. I the CHER database two Polsh paels are avalable: the frst oe referrg to years the secod to Orasa (999) aalyzes household welfare traectores durg the perod to detfy log-ter poverty ad detere the relevace of household asset edowets as deterats of household poverty ad vulerablty over te. Overall coe oblty Polad at that perod was hgh. I other words (9) Durato data aalyses beeft fro the use of dscrete-te odels. However avalable ecooetrc software s usually uable to accout for the saplg ethod used thereby rasg the probablty of saple selecto bas (Jes 995). Jes (995).

135 STATISTICS IN TRANSITION-ew seres Deceber the percetage of households that had chaged ther orgal posto was substatal. The treds repeated poverty durg the growth perod were slar so that durg the pre-trasto era: the fracto of households experecg two year poverty oscllated aroud 0 percet after t had worseed sgfcatly durg the recesso tself (Orasa 999). Ths paper s based o the secod pael referrg to I ths pael 08 households are corporated. Defg the category of poverty s a ey eleet assessg ts rage ad depth. The d of defto accepted wll detere whch groups of socety ay be acowledged as the ost vulerable to poverty rs. The choce of a ethod settg the poverty arg depeds o whether poverty s treated as a relatve or absolute category. I the frst case poverty s uderstood as a relatve deprvato the level of whch s depcted a relato to wealth of other better stuated ebers of the socety. I the secod case poverty s uderstood as a lac of fxed sources of coe depedet of the level of resources avalable to a overwhelg part of socety whch the povershed dvdual s a eber of. Kot (000 p. 8). Pae Podgórs Szulc (999).

136 86 N. Nehrebeca: Aalyss of teporary Table. A tred of real average equvalet coe as well as characterstcs of povershed households Poverty arg Characterstcs of povershed households ( PLN) Real average equvalet coe (for all households) ( %) % eda coe 50% eda coe Ipovershed households (%) Real average equvalet coe of poor fales ( PLN) Real average equvalet coe of wealthy fales ( PLN) Average poverty gap ( PLN) Icoe gap dex Ipovershed households (%) Real average equvalet coe of poor fales ( PLN) 0 Real average equvalet coe of wealthy fales ( PLN) Average poverty gap ( PLN) Icoe gap dex Source: CHER database: Polad ( ); ow calculatos. Lterature o the subect dscusses the followg les of poverty: absolute relatve subectve ad offcal. Nevertheless ost of the research shows the poverty s boudary as a proporto of cetral tedecy easures (average or eda) of a dstrbuto of a gve populato. I the poverty aalyss t s reasoable to use varous poverty boudares 5. I ths artcle I cosder a household (ad thus all ts ebers) as povershed 6 f ts real equvalet A poverty gap shows how far poor persos are fro the poverty arg that s how uch o average each faly lvg the poverty zoe should receve order to fd tself exactly o the poverty arg. A dex used to sythetcally assess a depth of poverty s a coe gap dex defed as: I N z N y z where: N a uber of households z a poverty arg y a coe of household equvalece scale of -th household (Pae Podgórs Szulc 999). Golowsa (997 p. 0-]. The les of absolute poverty were used research by Desczu Saewcz (995) Kurows (008) relatve les were used by GUS Polad (Kordos Ochoc 99) subectve les of poverty the followg studes (Kot Kasprzy 000). The relatve les of poverty were used studes by Eurostat ad by GUS Polad. I the Europea Uo coutres as a offcal boudary of poverty - half of a average coe - s used. 5 Steward Swaffed (999) Cappellar (000). 6 It s portat to otce that the level of poverty boudary ad ts proporto to deleatg the equvalet coe are a result of statstcal agreeet.

137 STATISTICS IN TRANSITION-ew seres Deceber coe s lower tha the poverty boudary le of 50% or 75% of real equvalet eda coe easured accordg to the OECD relatve scale (00/70/50). However t s portat to otce that accordg to Hageaars et al. (99) the OECD scale s too uch focused o large fales ad therefore the so-called OECD odfed scale has bee proposed. The choce of usg the OECD orgal scale research s ustfed for at least two reasos. Frstly the orgal scale s used by Eurostat ad secodly the 990s t showed a relatvely hgh coforty wth scales estated o the bass of real expeses of households tag part faly s budget research (Szulc 995). Table presets the prary characterstcs of poverty the aalyzed saple for Polad betwee Over ths perod the average coe decled by approxately %. The average real coe of the populato after a draatc decrease at the begg of the trasforato perod was cotuously creasg utl 998. Fro ths oet there was a vvd decle hgh over the last few years ecooc dyac followed by a later reducto of the rate of growth the real coe of the populato. The uber of dvduals havg coe below a relatve boudary of poverty of 50% eda coe after a crease 998 bega to drop. et at the relatve boudary of poverty of 75% eda coe the uber of povershed households shows a declg tred sce the begg of the studed perod. The aout eeded to cobat poverty wth the use of the poverty boudary of 50% eda coe s o average 00 PLN (about 700 PLN whe a poverty boudary of 75% eda coe s used). The average wealth of a povershed household group s lower by about 7% tha the poverty le ( 999 eve by %) wth the use of a poverty boudary of 50% eda coe (o average 8% whe a poverty boudary of 75% eda coe s appled)... Characterstcs of depedet varables used the study The choce of depedet varables the pael odel was based o prevous studes whch pertaed to poverty aalyss. Varables that have bee used the two types of the odels are as follows: poverty ext rates ad poverty retur rates are preseted below. They clude: I the study a et real equvalet coe was cosdered expressed year 000 currecy (dces for the followg years: [000] 07 [999] 07 8 [998] [997] ) (Iflato the years equalled: 9% 8% 7% Idces for prces of cosuer goods ad servces for ca be foud at Net equvalet coe was set usg a equvalet scale utlzed by GUS whch gves a weght of per head of the household ad 07 for every other adult perso (age>5 years) as well as 05 for every chld the household (age <5 years). Net equvalet coe was set usg a odfed equvalet scale whch gves a weght of per head of the household ad 05 for every other adult perso (age>5 years) as well as 0 for every chld the household (age <5 years).

138 88 N. Nehrebeca: Aalyss of teporary Durato of poverty ( years) - a varable used ext fro poverty odel: durato_poverty t (coposed of three levels: ext fro poverty after year ext fro poverty after years ext fro poverty after years). A varable used retur to poverty odel: durato of wealth ( years) durato_wealth t (coposed of three levels: retur to poverty after year retur to poverty after years retur to poverty after years). Geder of the head of the household: geder t (coposed of two types: ale feale). Professoal status of the head of the household: status t (coposed of four levels: eployed ad self-eployed retred ueployed ot actve professoally for other reasos). Educato status of the head of the household: educato t (coposed of three levels: coplete prary ad prary secodary ad vocatoal hgher). Place of lvg: cty coutrysde t (coposed of two types: cty coutrysde). Geographcal rego : rego t (coposed of four types: Easter Polad ad Wara ad Mazury Souther Polad Wester Polad ad Poeraa Cetre). The followg regos for the vovodshps: Easter Polad ad Wara ad Mazury: lubelse podarpace podlase warńso-azurse; Souther Polad: ałopolse opolse śląse śwętorzyse; Wester Polad ad Poeraa: dolośląse lubuse poorse zachodopoorse; Cetre: uawso-poorse łódze azowece welopolse. Ths classfcato s dfferet fro the dvso of the coutry to regos by GUS (GUS 008) however for the purpose of ths paper such a assupto was the ost coveet.

139 STATISTICS IN TRANSITION-ew seres Deceber Table. Characterstcs of depedet varables are used the odel Idepedet varables The rate of ext fro poverty Ipovershed fales Wealthy fales The rate of retur to poverty Ipovershed fales Wealthy fales Varables pertag to the head of the faly Geder (%): ale feale 8 5 Age (average years) Educato (%): coplete prary ad prary 8 secodary ad vocatoal hgher 08 Professoal status (%): eployed ad self-eployed retred ueployed ot actve for other reasos Extet of the balace the faly budget (%): surplus defct Varables pertag to the household Nuber of chldre below sx years of age (average) 0 Nuber of chldre of the age betwee of 6 ad 6 (average) 0 Place of eployet (%): cty 6 coutrysde 6 58 Rego (%): Easter Polad ad Wara ad Mazury Souther Polad 9 Wester Polad ad Poeraa Cetre Macroecooc varables Ueployet rate (average %) ear of etry to poverty (%): Nuber of households Source: CHER database: Polad ( ); ow calculatos.

140 90 N. Nehrebeca: Aalyss of teporary Extet of the balace the faly s budget: budget t (coposed of two types: surplus defct). ear of etry to poverty: year t (coposed of four levels: ). Age of the head of the household: age t ad age squared: age_ t (lterature o the subect ephaszes a olear relato betwee age ad ext fro ad retur to poverty). Nuber of chldre below 6 years of age: uber_chldre6 t. Nuber of chldre betwee the ages of 6 ad 6: uber_chldre6 t. Ueployet rate accordg to vovodshp: rate_ueployet t. For the correctess of coducted aalyss soe varables have bee trasfored to 0 varables. Table cludes characterstcs of these depedet varables used the odel.. Noparaetrc ethod Ths part presets the results of the study usg the oparaetrc ethod o the topc of durato of poverty. It also provdes dstrbutos of the years spet poverty oe-spell or ult-tes epsodes... Statstcs descrbg the dyac of poverty Table shows that whe usg a poverty arg of 50% eda coe 7% of the populato studed was affected by poverty (0% at a poverty arg of 75% eda coe) ths case % (86%) of the populato was cluded a prologed poverty. The expected aout of te spet poverty at a poverty arg of 50% eda coe for those eterg the pael ad studed durg Soe varables were trasfored to 0 varables: durato_poverty_0 t durato_poverty_0 t (f a faly exts the poverty after years years respectvely) durato_wealth_0 t durato_wealth_0 t ( f a household returs to poverty after years wealth after year wealth respectvely) geder_0 t ( f the head of the household s a feale) status_0 t status_0 t status_0 t ( f the head of the household s: retred ot eployed o actve for other reasos) educato_0 t educato_0 t ( f the head of the household has oe of the followg educato types: vocatoal ad secodary hgher) cty_coutrysde_0 t ( f the household s located the coutrysde) rego_0 t rego_0 t rego_0 t ( f the household s located as follows: Souther Polad Wester Polad the Cetre respectvely) budget_0 t ( f the faly had a egatve balace of the faly budget) year_0 t year_0 t year_0 t ( f the household etered poverty oe of the followg years respectvely). I the paraetrc ethod property arg of 75% eda coe has bee used (00/70/50 scale). The oparaetrc ethod provdes forato about a chage dvdual s behavor depedet o te uder assupto of oexstece of a partcular for of a evet dstrbuto (Frątcza Baber Gach-Cepela 005).

141 STATISTICS IN TRANSITION-ew seres Deceber the year perod equals to 0 part of the year; for the poverty arg of 75% eda coe t equals to 089 part of the year. Table. Household dstrbuto by the uber of years spet poverty Percetage share of povershed households wth the use of a relatve args Nuber of years of poverty poverty 50% eda 75% eda Source: CHER database: Polad ( ); ow calculatos. Table presets sequeces of coe years for the studed saple. If a household was poverty a gve year t s ared as U f ot the as N. The frst colu shows how log the poverty lasted whether the studed dvdual exted poverty ad whether they retured to t. A log-ter perspectve preseted Table ad Table ght be copared to a short-ter cross-secto vew proposed Table. A rate of poverty at a gve pot of te equals o average approxately % at a poverty arg of 75% eda coe ad about 8% at a poverty arg of 50% (Table ). Table shows though that 0% of the studed populato at the poverty arg of 75% eda coe (7% at a poverty arg of 50%) had bee affected by poverty at least oce. However the data Table deostrates that ay cases dvduals expereced poverty the subsequet years (ult-te epsodes). Oe ca thus reaso that households eterg (or extg) poverty ght be begg a log perod of reag that state ad what s ore portat they are o-dfferetable fro those that are located below the poverty arg oly oe or two observed years. As a result the extet of the pheoeo of poverty s uderestated. I each of these four years oe ca ote that ths apples to about 8% of povershed households at a poverty arg of 50% of the eda coe (% at a poverty arg of 75%) whle the etre fouryear perod shorter or loger epsode of poverty was overcoe by as uch as 7% of households at the poverty arg of 50% eda coe (0% at the poverty arg of 75%).

142 9 N. Nehrebeca: Aalyss of teporary Table. Household dstrbuto a -year sequece of states (U poverty N ot poverty) State sequeces Percetage share of povershed households usg a relatve args of poverty 50% eda 75% eda NNNN NNNU 5 NNUN 8 66 NNUU 8 6 NUNN 6 8 NUNU NUUN NUUU UNNN 0 70 UNNU 05 5 UNUN UNUU 0 UUNN UUNU 09 8 UUUN UUUU 8 86 Total Source: CHER database: Polad ( ); ow calculatos... Rates of ext fro ad re-etry to poverty (Kapla-Meer estator) I ths study of ext fro ad retur to poverty rates a oparaetrc ethod has bee used. Accordg to the Devcet s defto(00) the ext rates that are relevat ths cotext are the oes that refer to a cohort of dvduals ust fallg to poverty hece at rs of reag poverty thereafter. The re-etry rates refer stead to a cohort of dvduals ust startg a spell out of poverty ad so at rs of re-eterg. Ext rates are calculated by dvdg the uber of dvduals edg a spell after d years poverty by the total uber wth low coe for at least d years. The re-etry rates were calculated aalogously. To estate the ext rate the author used the followg sequeces: NUxx ad xnux where x N U (NNUN NUNN NUNU UNUN NNUU NUUN UNUU NUUU) however the retur rate: UNxx ad xunx where x N U (UNUU UNUN UUNU NUNU UUNN NUNN UNNU UNNN). Ule the sple calculato of the uber of years poverty such a perodcal approach ca clude the rght-sde cesored observatos. For the purposes of ths aalyss the Devcet (00 p. 8 9).

143 STATISTICS IN TRANSITION-ew seres Deceber left-sde cesored observatos have bee excluded o accout of whch the research study starts fro 998 or later. Table 5. Survval fucto ad rates of ext fro poverty (Kapla-Meer estator) Relatve poverty arg ears 50% eda coe 75% eda coe Survval fucto Retur rate Survval fucto Retur rate (-) 066 (009) (-) 059 (000) 065 (0099) 008 (009) 078 (008) 0 (00608) 06 (00) (0058) - *) bracets stadard errors are provded. Source: CHER database: Polad ( ); ow calculatos. Table 5 shows ext rates fro poverty usg a relatve poverty arg. A estated hazard rate cofrs egatve effects of poverty durato: the loger the dvdual lves poverty the lesser probablty that ths state wll chage the ext perod. For the cohort of dvduals that beg a perod of poverty the probablty of ext after the frst year s equal to about 6% ad after two years to approxately % (a relatve poverty arg of 50% eda coe) however for the sae cohort the probablty of ext fro poverty after the frst year s equal to about 5% ad after two years to approxately % (a relatve poverty arg of 75% eda coe). Table 6. Survval fucto ad rates of re-etry to poverty (Kapla-Meer estator) Relatve poverty arg ears 50% eda coe 75% eda coe Survval fucto Retur rate Survval fucto Retur rate (-) 09 (00) (-) 0008 (000) 0757 (007) 0860 (0065) 0699 (000) 0099 (00) 066 (0088) (0065) - *) bracets stadard errors are provded. Source: CHER database: Polad ( ); ow calculatos. Geeralzg results fro Table 5 ad Table 6 ples the thess that low coe cludes a wde spectru of households. et t s ot a fxed group. Although there are dvduals preset t who are peraetly poor ay fales ext fro ad eter poverty.

144 9 N. Nehrebeca: Aalyss of teporary.. Poverty durato oe-spell ad ult-te epsodes The estato of ext fro ad re-etry to poverty allows us to troduce dstrbuto of te spet poverty. Such dstrbuto s a basc easure of durato of poverty. The dstrbuto of years spet oe-spell epsodes of poverty has bee calculated usg oly ext rates preseted Table 5 (e.g. two years spet poverty depcted as (NUUN) where N - a perod of wealth ad U - a perod of poverty). It was assued that e(d) ad r(d) are respectvely ext rate fro ad reetry to poverty after d years. Wth excepto of the left-sde cesored perods the probablty of ths sequece s calculated as: (-e())*e(). The dstrbuto of years spet poverty the case of ult-te epsodes was calculated usg the ext ad re-etry rates preseted Table 5 ad Table 6 (e.g. two years poverty depcted as (NUNU)). Wth excepto of the left-sde cesored perods the probablty of ths sequece s calculated as: (UNU)e()*r(). To calculate the probablty of observg two years of lvg poverty wth these four years oe ust calculate the probablty of occurrece of all possble cobatos whch geerate a su of two years poverty ad addg the up. I coparso to the forecasts for oe-spell ad ult-te perods also a dstrbuto of te spet poverty was calculated usg the followg sequece: (N U x x) where x N U. Coparg Colus ad Table 7 oe ca otce that oe-spell epsodes the dstrbuto of te spet poverty was overestated for oe year spet poverty. For two-year durato of poverty the use of oe-spell epsode approach each case results uderestato of dstrbuto of the te spet poverty. Table 7. Dstrbuto of years spet poverty for the cohort of dvduals begg a perod poverty 998 ears poverty Dstrbuto oe-spell epsodes 50% eda coe 75% eda coe Dstrbuto of years spet poverty the followg three years expected actual 50% eda 75% eda 50% eda 75% eda coe coe coe coe Source: CHER database: Polad ( ); ow calculatos. Expected dstrbuto of years spet poverty was deleated usg ext ad etry rates fro/to poverty tae fro the Kapla-Maer odel (Table 5 ad Table 6). Actual dstrbuto of years spet poverty was deleated usg the followg sequece (NUxx) where xnu.

145 STATISTICS IN TRANSITION-ew seres Deceber It s portat to ephasze that the above etoed aalyss assues that all observed epsodes perta to a hoogeeous populato. However t s ore probable that dfferet fales havg specal characterstcs (observed ad uobserved) eet varous rates of ext ad re-etry to poverty whch explas why peraetly poor households exst. As a result oe ust chage fro ulateral to ultlateral approach whch allows for depedece of a ext rates ad re-etry rates o portat soco-ecooc correlatos. I the paraetrc ethod the poverty arg of 75% eda coe s used because t gves ore possbltes to study the processes of ext fro ad re-etry to poverty a shortru.. Paraetrc ethod: odel wth dscrete te tag to cosderato observed ad uobserved heterogeety of dvduals Ths part presets the results of studes o durato of poverty obtaed by the use of paraetrc ethod.the estated odel cotas varables pertag to characterstcs of the head of a household as well as the household tself ad also codtos o the labor aret whch have pact o the probablty of ext ad re-etry to poverty later o.. The estato of the two copleetary log-log odels wth dscrete te has bee coducted. The frst oe (Model I) cludes elastc o-paraetrc specfcato for the base hazard fucto. The secod oe (Model II) descrbes the base hazard usg the ult-desoal durato fucto that s: θ 0( t) atbt ct. The use of these two odels was ustfed by the observatos ade by Meyer (990) that: paraeters of the base hazard fucto depct a portat characterstc of the data whch would be otted f the odel would have bee estated by a sple paraetrc base hazard fucto. Models Ia ad IIa that clude a uobserved heterogeety have bee also estated... Who exts fro poverty? Table 8 presets the aalyss of durato of epsodes of poverty that ed by the household gag a wealth status. Model cofrs a egatve relato betwee durato of poverty ad the probablty of leavg these codtos whch was earler observed o the bass of the results fro Table 5: the loger a faly reas poverty the harder t s for the to re-eter the state of wealth. The probablty of ext rate fro poverty also depeds o the characterstcs of the head of a faly (e.g. geder educato) ad a household (e.g. uber of chldre faly s aual budget). Basg o collected evdece households About paraetrc odels t s sad that the aalytcal for of probablty dstrbuto of desty s ow.

146 96 N. Nehrebeca: Aalyss of teporary headed by woe rea poverty loger whch s show by Model II. The probablty of ext fro poverty decreases alog wth the crease the uber of chldre a household (aged up to 6 years) as bearg chldre aes fullte eployet ore dffcult (especally the case of woe). Moreover the results of the estato show that the hgher educatoal level of the head of a faly the lower the chace of experecg a relatvely log perod of poverty by ther faly ad the easer fght wth poverty. Havg a hgher educato gves a ay-tes greater chace of ext fro poverty tha the case where the head of a faly has a secodary or vocatoal educato. A portat aspect of the possblty to ext poverty s also the status of the head of a faly o the labor aret. Households where the faly s headed by a ueployed perso are a decsvely worse stuato. Furtherore % crease a ueployet rate a vovodshp decreases the probablty of ext fro poverty by approxately 5 % accordg to Model I. A faly budget defct has also a egatve pact o ext fro poverty aother factor deleatg durato of poverty beg a acroecooc stuato of a coutry as well as a perod o whch the begg of povershet fell (for household whch becae poor ad 000 a chace to ext poverty was uch hgher that for those the base group whch etered poverty 997). Ths pheoeo could have tae place due to ecooc growth Polad. Results of teratoal research (Barro 999) ad terdepedeces betwee uequal geeral coe ad ecooc growth Polad durg the trasforato perod cofr that oe caot expect the codtos of Polsh ecooy a sgfcat ltato of coe dsproportos. A observatos of aual fluctuato of GDP crease rates ad G factor over the years cofr that Polad a hgher speed of ecooc growth was accopaed by a lower buld up of equaltes ters of socal dvso of coe(jabłońs 00).

147 STATISTICS IN TRANSITION-ew seres Deceber Table 8. Aalyss of poverty durato Polad ext fro poverty Idepedet varables Durato of poverty: Wthout uobserved heterogeety Wth uobserved heterogeety Model I Model II Model Ia Model IIa Paraetr [stadard error] two years - 090* [0095] * [0] - three years - 989* [0097] * [06] - ear of etry to poverty: * [05] - 9* [09] * [0] - 56* [06] * [0] - 77* [070] - Varables pertag to the head of the faly Geder (feale) * [0087] - - 0* [05] Age - 006* [000] - - 0* [00] - Age_/00 009* [00] * [00] - Professoal status: retred - 00 [069] - 079* [0] [00] - 057** [09] ueployed * [06] - 70* [08] - 098** [056] - 005* [05] ot actve professoally - 069* [009] * [00] - 08* [05] - 9* [07] Educato: vocatoal or secodary - 00 [008] - 0 [09] hgher - 65* [07] - 098* [060] Varables pertag to the household Nuber of chldre up to 6 years - 06* [0060] - 095* [0055] - 07* [0075] - 067* [0] Nuber of chldre betwee 6 ad 6 years of age - 00* [000] - 005* [009] - 05* [0050] - 059* [007] Budget defct of the faly * [008] - 07* [0078] * [00] * [09] Ueployet rate - 009** [00] - 009* [000] -008* [008] [009] t * [095] - - 5* [07] t - 97* [055] - 9* [0] t - - 0* [00] * [006] Nuber of fales 97 Logarth lelhood Stars sgfy a sgfcat of paraeters of the followg levels: * % ** 5%*** 0%. Source: CHER database: Polad ( ); ow calculatos. The results of the estatos of Models Ia ad IIa tag to cosderato uobserved heterogeety of the studed dvduals are cluded Table 8. These dffer fro the results obtaed o the bass of Models I ad II. I soe cases a absolute value of coeffcets s greater whch stregthes the pact of regressors o a chace of ext fro poverty. Models Ia ad IIa are also characterzed by a greater value of the logarth lelhood.

148 98 N. Nehrebeca: Aalyss of teporary.. Who returs to poverty? The results of the estated odels of the chaces of the fales to retur to poverty are cluded Table 9. It s portat to otce that estated coeffcets are characterzed by a greater varablty tha the case of ext rates. Model I shows a postve terdepedece betwee durato wealth after ext fro poverty ad re-etry to t whch weaes wth a crease years spet wealth. However o the bass of the odel cludg heterogeety of the studed dvduals oe ca dscer that eve after two years spet wealth there s a egatve relatoshp betwee wealth ad re-eterg poverty. Therefore the loger a perso stays uaffected by poverty the lower the chace of re-eterg t. Moreover the probablty of returg to poverty decreases f: fales lve the cty the head of a faly s a a (a terestg fact s that ths varable Model I for estatg the rate of ext fro poverty was sgfcat) ad whe the educato level of the head of a faly creases. Model II also cludes varables descrbg a geographcal rego where the household s stuated. The results of estatos do ot show sgfcat dffereces betwee vovodshps ( each case there was a egatve relato). Households vovodshps of Wester Polad are however a better stuato coparso to households vovodshps of the Easter Polad as well as Wara ad Mazury.

149 STATISTICS IN TRANSITION-ew seres Deceber Table 9. Aalyss of poverty durato Polad retur to poverty Idepedet varables Durato of wealth: Wthout uobserved heterogeety Wth uobserved heterogeety Model I Model II Model Ia Paraetr [stadard error] two years 056** [09] [050] oe year 6* [087] - 65* [09] Varables pertag to the head of the faly Geder (ale) - 096* [06] * [00] - 055** [078] Age_/00-00* [0005] - 005* [0006] - 00* [0007] Professoal status: ueployed 9* [0] 97* [06] 080*** [06] Educato: vocatoal or secodary -067* [05] - 055* [05] ** [0] hgher -79** [077] - 00* [070] - 57** [075] Varables pertag to the household Budget defct of the faly 0* [0089] - 8* [0098] Rego: South * [00] - Easter * [0] - Cetre * [066] - Place of lvg: coutrysde - 087** [06] - - 0** [055] t ** [055] t - 086* [007] Nuber of fales 0 Logarth lelhood Stars sgfy a sgfcat of paraeters of the followg levels: * % ** 5%*** 0%. Source: CHER database: Polad ( ); ow calculatos. Coclusos Poverty has always bee oe of the ost portat probles of the coteporary Polad. The research based o pael data represetatve of the whole coutry has show a low coe dyac of fales lvg Polad betwee The total uber of years spet poverty ad varous sequeces of etry to ad ext fro poverty have dcated the peraece of ths pheoeo the populato. As follows fro the coducted aalyss a sall uber of fales (less tha 5% of the populato at a poverty arg of 50% of eda coe ad less tha 9% at a poverty arg equal to 75%) was povershed for the etre perod of the study yet a uch hgher percetage of the etre populato expereced

150 500 N. Nehrebeca: Aalyss of teporary poverty at least oce (7% at the poverty arg of 50% of eda coe ad 0% at the poverty arg equal to 75%). A relatvely wde array of households lve o low coe. However t s ot a fxed group. Eve though there are soe dvduals who are peraetly povershed poverty trastos occur. Approxately 65% of the studed fales at a poverty arg of 50% (5% at the poverty arg equal to 75%) coe out of poverty after the frst year of beg povershed. Nevertheless oly 5% at a poverty arg of 50% (0% at the poverty arg of 75%) becoe poor oce aga. After two years of lvg poverty 0% of fales ext t but 8% at a poverty arg equal to 50% (0% at a poverty arg equal to 75%) re-eter a group of povershed fales. Because of a short te perod of the saple the study could ot have corporated a wder aalyss of ths type. Oe ca assue however. that the loger the faly lves poverty the ore dffcult t s for the to chage ther stuato ad eve f ther coe creases above the poverty arg the probablty of these households becog poor oce aga s stll qute hgh. Aother portat ssue to aalyze s the dstrbuto of the uber of years spet poverty cosderg exts ad re-etres of fales to ths stuato. Tag to accout the sequeces of epsodes of beg povershed ad wealthy where the fales foud theselves partcular years oe ca forecast that as uch as 50% of the studed fales at a poverty arg equal to 50% (65% at a poverty arg equal 75%) wll spet at least followg years poverty. However whe cludg oly oe-spell epsodes of becog povershed these progoses coe to a level of ca.5% of the studed fales at a poverty arg equal to 50% (8% at a poverty arg equal to 75%). The actual values aout to 8% at a poverty arg of 50% (60% at a poverty arg equal to75%) therefore cludg the sequeces of the epsodes proves the results. The observed results also show that the legth of te of poverty ay deped o a type of the defto of ext fro ad re-etry to poverty rates that s accepted. The aalyss usg the paraetrc ethod llustrates that there are groups of populato ore vulerable to droppg below the poverty arg wth hgher probablty of reag povershed over a loger perod of te. These are households coposed of a larger uber of ot oly chldre but also adults where the head of a faly s a older perso (aly a woa) wth a low educato level. The lfe vovodshps wth a hgh rate of ueployet s a factor creasg poverty ad especally edagered are households where the head of a faly s ueployed. Orasa (999) used four-year pael data (99 999) fro Polad s Household Budget Survey to explore the dstcto betwee trastory ad logter poverty ad exae poverty oblty. The secto of populato that could ze or avod chroc poverty Polad cluded those lvg urba areas headed by older ad better educated wth few chldre ad ueployed ebers ad possessg facal or physcal assets. Households wth a larger shp etwor faced sgfcatly less dager of fallg to chroc poverty or vulerablty.

151 STATISTICS IN TRANSITION-ew seres Deceber The results of the estato of odels cludg uobserved heterogeety of studed dvduals cofr a egatve terdepedece betwee the ext fro poverty rate ad ts durato. For parets stayg a relatvely log perods of te poverty t s uch ore dffcult to ext t o ther ow. Though the loger the household that exted poverty reas outsde that stuato the lower the chace that they wll retur (a egatve relato exsts after two years of beg above the poverty arg). The above coclusos ay be useful for preparg plas to deal wth logter poverty Polad. They wll allow for a better uderstadg of the poverty pheoeo Polad ad factors that cause t. The results obtaed ght be helpful for a effectve polcy forulato: addg to a coe of eployed household ebers early detfcato of fales ofte eterg the sphere of poverty ad peraetly povershed odelg of codtos o the labor aret so as to lower the cdece of poverty Polad. REFERENCES ALLISON P. D. (98) Dscrete-te ethods for the aalyss of evet hstores Socologcal Methodology 98 (S. Lehardt ed.) Jossey-Bass Publshers Sa Fracsco 98. BANE M. ELLWOOD D. (986) Slppg Ito ad Out of Poverty: The Dyacs of Spells Joural of Hua Resources Vol.. BARRO R.. J. (999) Iequalty growth ad vestet Worg Paper 708 Natoal Bureau of Ecooc Research. BIEWEN M. (00) Who are the chroc poor? Evdece o the Extet ad the Coposto of Chroc Poverty Geray IZA Dscusso Paper No CIURA G. (00) Poverty ad povershed area The Bureau of Research No. 88 (I Polsh). CAPPELLARI L. JENKINS S. (00) Modelg low coe trasto Dscusso Papers of DIW Berl fro DIW Berl Gera Isttute for Ecooc Research No. 88. CIECIELĄG J. TOMASZEWSKI A. (00) Ecooetrc aalyss of pael data (I Polsh) Warsaw 00. DEVICIENTI F. (00) Poverty persstece Brta: a ultvarate aalyss usg the BHPS Joural Of Ecoocs No. 9. DEVICIENTI F. (00) Estatg poverty persstece Brta LABORatoro R. Revell Worg Papers Seres.

152 50 N. Nehrebeca: Aalyss of teporary DOLTON P. VAN DER KLAAUW W. (995) Leavg teachg the UK: a durato aalyss Ecooc Joural Vol. 05 No. 9. DUNCAN G. (999) The PSID ad Me IPR worg papers. FRĄTCZAK E. BABIKER H. GACH-CIEPIELA U. (005) Hstory occuraces aalyss Eleets theory chose practcal exaples (I Polsh) Warsaw 005. Cetral Statstcal Offce Exstece stuato of households 00 (I Polsh) GOLINOWSKA S. (996) Polsh poverty. Crtera. Assesset. Couteracto. The Isttute of Labour ad Socal Studes (I Polsh) Warsaw 996. GOLINOWSKA S. (997) Polsh poverty II. Crtera. Assesset. Couteracto. The Isttute of Labour ad Socal Studes (I Polsh) Warsaw 997. HANSEN J. WALHBERG R. (00) Poverty persstece Swede IZA Dscusso Paper No. 09. JABŁOŃSKI Ł. (00) Ecooc growth or ltg coe equalty Polad? No (I Polsh) Rzeszów 00. JARVIS S. JENKINS S. (997) Low coe dyacs 990s Brta Fscal Studes No. 8. JENKINS S. (995) Easy estato ethod for dscrete-te durato odels Oxford Bullet of Ecoocs ad Statstcs No. 57. JENKINS S. (997) Estato of dscrete te proportoal hazards odels Stata Techcal Bullet Reprts. JENKINS S. (00) Survval Aalyss upublshed auscrpt Isttute for Socal ad Ecooc. KASPRZK B. (000) Methodologcal aspects of assessg the level of wealth Cracow Uversty of Ecoocs (I Polsh) Cracow 000. KIEFER N. (988) Ecooc Durato Data ad Hazard Fuctos Joural Of Ecooc Lterature Vol. VI. KORDOS J. OCHOCKI A. (99) Probles wth easurg poverty EWG coutres ad Polad Statstcal News (I Polsh) Warsaw 99. KOT S. M. (995) Modellg level of wealth. Theory ad applcato Ossoleu (I Polsh) Wroclaw 995. KOT S. M. (998) The Cracow Poverty Le Cracow Uversty of Ecoocs (I Polsh) Cracow 998.

153 STATISTICS IN TRANSITION-ew seres Deceber KUMOR P. SZTAUDNGER J. (007) Optal copesato dfferetato Polad ecooetrc aalyss Ecoost No. (I Polsh) Warsaw 007. KUROWSKI P. (007) Study of the odfed u exstace level 006 The Isttute of Labour ad Socal Studes (I Polsh) KUROWSKI P. (008) Study of the level ad structure of the odfed u exstace level 007 The Isttute of Labour ad Socal Studes (I Polsh) LILLARD L. A. WILLIS R. J. (978) Dyacs aspects of earg oblty Ecooetrca Vol. 6. MEER B. D. (990) Ueployet surace ad ueployet spell Ecooetrca Vol. 58. OKRASA W. (999) Who Avods ad Who Escapes fro Poverty durg the Trasto? Evdece fro Polsh Pael Data The World Ba Polcy Research Worg Paper 8 Washgto. PANEK T. PODGÓRSKI J. SZULC A. (999) Poverty: theory ad practcal assesset Warsaw School of Ecoocs (I Polsh) Warsaw 999. PANEK T. (007) Poverty ad equalty Socal Statstcs Polsh Ecoocs Publshers (I Polsh) Warsaw 007. PRENTICE R. GLOECKER L. (978) Regresso aalyss of grouped survval data wth applcato to breast cacer data Boetrcs Vol.. Vovodshp aual statstcal yearboo Cetral Statstcal Offce (I Polsh) Warsaw RADZIUKIEWICZ M. (007) Poverty reach Polad (I Polsh) Polsh Ecoocs Publshers 007. RUSNAK Z. KOŚN M. (00) Ipact of chages equvalece scales o the coe ad equvalet expese dstrbutos : Applcatos of Statstcs ad Matheatcs Ecoocs Stasewcz W. (Eds.) Wrocław Uversty of Ecoocs (I Polsh) Wroclaw 00. SHORROCKS A. K. (978) The Measureet of Moblty. Ecooetrca Vol. 6 No. 5 p STANISZ A. (00) Practcal edce 00/0 Practcal edce (I Polsh). STEVENS A. (999) Clbg out of poverty fallg bac. Measurg the persstece of poverty over ultple spells Joural of Hua Resources Vol. No..

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155 STATISTICS IN TRANSITION-ew seres Deceber STATISTICS IN TRANSITION-ew seres Deceber 009 Vol. 0 No. pp APPLING THE HADAMARD PRODUCT TO DECOMPOSE GINI CONCENTRATION REDISTRIBUTION AND RE-RANKING INDEES Achlle Verzz * ABSTRACT G ad cocetrato dexes are well ow useful tools aalysg redstrbuto ad re-rag effects of taxes wth respect to a populato of coe earers. There are several attepts the lterature to decopose G ad re-rag dces to aalyse potetal redstrbuto effects ad the ufaress of a tax systes cludg oes that cosder cotguous coe groups beg created by dvdg the pre-tax coe parade accordg to the sae badwdth. However earers ay be very ofte splt to groups characterzed by socal ad deographc aspects or by other characterstcs: these crcustaces groups ca easly overlap. I ths paper we cosder a ore geeral stuato that taes to accout overlappg aog groups; we obta atrx copact fors for G ad cocetrato dexes ad cosequetly for redstrbuto ad re-rag dexes. I dervg forulae the so called atrx Hadaard product s extesvely used. Matrx algebra allows to wrte dexes algg coes a o decreasg order ether wth respect to post-tax coe or to pre-tax coes. Moreover atrx copact forulae allow a orgal dscusso for the sgs of the wth group across group betwee ad trasvarato copoets to whch the Atso-Plotc-Kaway rerag dex ca splt. Key words: G ad cocetrato dexes decopostos Tax redstrbutve effects Tax re-rag effects Hadaard product. * Uverstà degl Stud d Mlao DEAS [email protected]. Ths paper s part of a research proect ot wth Mara Govaa Mot (Uverstà degl Stud d Mlao ad Uverstà degl Stud d Mlao-Bcocca) ad Mauro Muss (Uverstà degl Stud d Mlao-Bcocca). The author thas hs research fellows for geeral dscussos observatos ad revsos. Moreover the author expresses specal grattude to Maro Falva for havg ecouraged the atrx approach adopted the preset paper ad for all hs precous suggestos ad to Gorgo Pederzol for hs precse ad helpful coets. Reag defceces ad staes are exclusvely due to author s resposblty.

156 506 A. Verzz: Applyg the Hadaard Itroducto It s ow that dealg wth a trasferable pheoeo where uts are classfable to groups G dex fals to decopose addtvely to a betwee ad a wth copoet f the group rages overlap. Followg Bahattacharya ad Mahalaobs (967) a uber of G decopostos was proposed (Rao (969) Pyatt (976) Mooheree ad Shorrocs (98) Slber (989) tzha ad Lera (99) Labert ad Aroso (99) tzha (99) Dagu (997)) ad after Labert ad Aroso (99) the thrd copoet of the covetoal G dex decoposto s deoted by overlappg ter. Mot (007) shows that the covetoal ad the Dagu (997) decoposto are detcal so that a alteratve way to calculate the overlappg ter ca be derved fro the decoposto suggested by ths author. Aroso Johso ad Labert (99) Urba ad Labert (008) use G ad cocetrato dex decoposto to detfy ad evaluate potetal dstrbutve effects ad ufaress a tax syste. These authors cosder cotguous coe groups created by dvdg the pre-tax coe parade accordg to a detcal badwdth so that the pre-tax coe parade excludes overlappg by costructo. I the preset paper we cosder coes gathered to groups characterzed by socal deographc or coe sources characterstcs so that overlappg aog groups eed ot to be excluded. Our results are obtaed usg the G dex decoposto derved fro Dagu decoposto (Mot ad Satoro 007 Mot 008). Mag use of the Hadaard product the frst secto we preset G ad cocetrato dexes copact atrx fors. I the secod secto we troduce groups preset G ad cocetrato dexes ad show how wth groups across betwee groups ad trasvarato copoets ca be wrtte atrx copact fors. Ls fro atrx copact fors ad scalar fors are reported: soe scalar expressos are well ow lterature whle others appears as odfcatos of already well ow fors. Secto presets atrx fors for redstrbuto ad re-rag dexes together wth ther wth across betwee groups ad trasvarato copoets. I the fourth secto we show how the sgs of Atso-Plotc-Kawa (Plotc 98) re-rag dex copoets ca be aalysed thas to the algebrac tools preseted the paper.. Matrx fors for cocetrato ad G dexes Let ad be two real o egatve statstcal varables that descrbe a trasferable pheoeo for a populato of K uts K N. I ths paper we suppose that represets coe before taxato ad after-tax coe; ot

157 STATISTICS IN TRANSITION-ew seres Deceber frequetly the par ( x y ) has assocated a weght p ( K) K p N. Furtherore easurg cocetrato we geerally eed to ra ether x or y a o-decreasg order: whe the eleets are raed a o-decreasg x y p ; order the sequece of ( x y p ) trplets wll be dcated as {( )} aalogously {( x y p )} wll deote the sequece of ( ) eleets are raed a o-decreasg order. The cocetrato dex for the orderg {( )} x y p whe the x y p s defed as K yp K yp p C ( y y) pp μn μn N μn K K ( y y) ppi μ N () : > 0 I 0: 0 : < 0 where μ s the weghed ea of the observatos o. Obvously the x y p the cocetrato dex C cocdes wth the G dex orderg {( ) } G ad aalogously the orderg {( )} x y p C G. Geerally whe tax effects are aalyzed oe cosders the G dex for the pre-tax dstrbuto G the G dex for the post-tax dstrbuto G ad the cocetrato dex for the post tax dstrbutos C wth coes raed accordg to the {( )} x y p orderg. The author s debt wth Mara Mot for the suggesto to express the cocetrato dex by dffereces betwee coes: ths suggesto s at the bass of ths paper. It ca be show that expressos () the frst forula s equal to the secod oe: the proof ca be easly obtaed followg the deostrato that Ladea (99 Ch...) gves for the G dex. I the rght had sde of () the frst copoet calculates the oralzed cocetrato. I the case where the y s are a o decreasg order the secod oe s the oralzed ea absolute dfferece that s ( μ ) K K. G N y y p p Δ μ The dcator fucto I s a partcular case of geeralzed fuctos cosdered Falva (000): ths artcle ca be cosulted for I propertes. For deftos cocerg cocetrato dexes ad ther relatos wth G dexes see e.g. Kawa (980) partcular Ch. 5 ad 8.

158 508 A. Verzz: Applyg the Hadaard I order to pass to a atrx represetato we stac the K observatos o ad the weghts P to K vectors: whe referrg to the orderg x y p the vectors wll be dcated as x y ad p whle referrg to the {( ) } orderg {( x y p )} the vectors wll be labelled as x y ad p that s whe eleets a vector are raed a o-decreasg order o label wll be added coversely whe they are ordered accordg to a o-decreasg order for aother varable ths varable wll be explctly dcated. We also troduce the followg deftos: S s wll deote a K K se-syetrc atrx wth dagoal eleets equal to zero super-dagoal eleets equal to ad sub-dagoal eleets equal to ; for a K vector that has etres equal to ; D ad D wll deote the K K atrces D ( x' x' ) D ( y' y' ). The by ag use of the Hadaard product we ca express the dexes G ad G as follows : G where μ ad respectvely. p's ( D ) p G ( ) μ N μ N p ' S D p () μ are the weghed ea of the observatos o ad o I addto by troducg the K K atrx ( ) the cocetrato dex copact for as C ( ) μ N D y ' y ' we ca wrte p ' S D p () The trasforato fro vectors y ad p to vectors y ad p ca be perfored by a proper K K perutato atrx E. The reverse trasforato fro y ad p to y ad p ca be obtaed through the atrx E whch s equal to E'. Forally The Hadaard product for two atrces A ad B s defed f both of the have the sae uber of rows ad the sae uber of colus: a b a b. For the defto ad propertes of the Hadaard product see e.g. Falva (98 Appedx) ad (987 Ch. ) Schott (005 Ch. 5).

159 STATISTICS IN TRANSITION-ew seres Deceber y Ey ye'y x Ex x E'x p Ep p E'p () We shall show that wth soe sutable algebrac perutatos of the eleets of S t s possble to reforulate both the atrces D ad the vectors p () ad x y p orderg () accordg ether to the {( x y p )} or to the {( )} atag both G ad cocetrato dexes uchaged. Ths leads to rewrte the expressos of forula () as G or as G ( ) p ' ESE' D p ad G ( ) μ N p ' S D p (5) μ N p's ( D ) p ad G ( ) μ N where D ( y ' y ' ) ad ( ) holds: p'e'se D p (6) μ N D x ' x ' respectvely. Moreover C ca be gve the followg alteratve for: Proof Cosder C p'e'se D p (7) ( ) μ N G as specfed () ad (6). As EE' E'E I the followg ( ) ( ) ( ) p ' S D p p 'EE' S D EE'p p ' E'SE E'D E p by eepg d the oteworthy property of the Hadaard product E' S D E E'SE E'D E (Falva 996 property v page. 57). ( ) ( ) ( ) Notcg that E'D E E' ( x' x' ) E ( x'e E'x' ) ( ) x ' x' D as E' ad 'E ' the equvalece of expresso () ad expresso (6) for G s proved. The equvalece of expressos () ad (5) for G ca be lewse proved. Ideed the followg holds: p ' S D p p 'E'E S D E'Ep p ' ESE' D p ( ) ( ) ( ) upo otcg that EDE' y'e' Ey' y ' y' D.

160 50 A. Verzz: Applyg the Hadaard As far as C s cocered expresso () turs out to be equvalet to expresso (7) upo otcg that E'D E y 'E E'y ' y' y' D.. Itroducg groups A populato of coe earers ca be parttoed to H groups H N whch ca be characterzed by coe sources or by socal ad deographc aspects: typcal group characterzatos are faly coposto depedet/odepedet worer e/woe geographc area etc. Dagu (997) decoposes the G coeffcet to wth groups (heceforth W) ad a across groups (heceforth AG) copoet. Dagu calls ths latter copoet gross betwee). W AG Hece G G G. I addto Dagu splts the AG copoet to a betwee ad a trasvarato copoet: G G G. The betwee AG B T B copoet G s the G (weghed) dex whch results whe all values wth the sae group are replaced by ther (weghed) average; the trasvarato T copoet G easures the overlappg aog groups: t s zero whe o AG overlappg exsts ad t s equal to G whe all group averages are equal. Extedg Dagu s decopostos to cocetrato dexes we ca splt C W AG to the two copoets W ad AG ad wrte C C C accordgly wth C ( y y ) p p I I (8) K K W h μ N C ( y y ) p p ( I ) I (9) K K AG h μ N I I (8) ad (9) I s as defed () above ad h s a dcator I h fucto: yad y f both belog to the sae group h (h...h) I h 0 yad y f do ot. For ore detals o the expresso of the G copoets the Dagu decoposto see e.g. Mot (008).

161 STATISTICS IN TRANSITION-ew seres Deceber C AG W Slar expressos hold for C ( ) W AG AG W G C G ad C W G AG G. I partcular for what cocers G W ad G AG the product y y I ca be replaced by the absolute dfferece y y. W AG I order to foralze copact atrx fors for C ad C t s worth to troduce a proper otato. More precsely J wll deote a K K atrx wth H all eleets equal to oe W w hw h' a K K atrx the {( )} h x y p orderg where w h stads for a K vector wth the -th etry equal to oe f the coe the -th posto belogs to group h (h..h) whereas t s zero otherwse. The atrx W whe appled to S D expresso () allows to detect the group fro the whole K s ( y y) atrx ( ) H Kh dffereces belogg to the sae h coe dffereces. Coversely the H J W whe appled to S D allows to detect the K Kh h dffereces betwee coes belogg to dfferet groups. Cosder ow the followg expressos for the W ad AG copoets of C : G C C p ' W S D p (0) ( ) W μ N p ' ( J W ) S D p () AG μ N W AG It s edate to verfy that C C C. Slar expressos for W W AG AG C ad for G C wth G W ad W W AG AG C ad G C ca be obtaed by substtutg p wth p W D wth D. Lewse the correspodg expressos for are obtaed by replacg μ wth D. Observe also that W E'WE ad W EWE'. μ ad D wth Moreover Dagu (997) splts G AG to the copoets G B ad G T brgg subdvso to the fore. Let uss ow label each subect trplet of observatos o ad P by a par of dexes (h) stead of oe as before: h refers to the group (h.h) whereas (.K h ) refers to the posto w h Ew h ad h h w E'w.

162 5 A. Verzz: Applyg the Hadaard that the subect occupes wth the h-th group; ote that ph Nh ad K h H H ph Nh N. h h Dagu s represetatos are: H H K h K g G y y p p h g h g μ N h g K h () G y y p p () H K K h h W μ N h h h h h G y y p p H H K h K g AG h g h g μ g h N h g () G p p p p H H K h K g H H B h g h g h g h g N μ μ μ h g μn μ μ h g (5a) where μ h represets the coe average of the h-th group (h...h). G y y p p K h Kg B h g h g μ N h g H h ( ) (5b) H h Kg Kh T G y h yg ph pg μ N h g { yh < yg } K h where p p ad pg pg. h h (6) K g We refer to Mot ad Satoro (007) forula (6) partcular for the dervato of expresso (5b). Expressos () () () ad the frst ter o the rght had sde (5a) do ot eed rag values; whereas (5b) ad (6) eed groups to be raed accordg to ther averages. Let us ow order the values (ad the related P ad possbly values) so that () wth each group they are raed a o-decreasg order; () groups are alged a o-decreasg order wth respect to the ther averages. The the values parade becoes ( y y... y )...( y y... y )...( y y... y ) y' A K h h h K H H H K (7) h H

163 STATISTICS IN TRANSITION-ew seres Deceber yh y h ( K h ) ad μh μ h (h.h). We shall deote the orderg gve by (7) as the {( )} The {( )} A x y p orderg. A x y p orderg ca be troduced lewse: accordg to ths orderg the values together wth the related ad P values are dstrbuted to the H groups such that () wth each group the x s are raed a o-decreasg order; () groups are a o-decreasg order wth respect to ther averages. x y p orderg wll appear as Thus for what cocers the values the {( )} A ( x x... x )...( x x... x )...( x x... x ) x' (8) h H A K h h h K H H H K xh x h ( K h ) ad μh μ h (h.h). The vectors y A (7) ad x A (8) ca be expressed as fuctos of y ad x respectvely by troducg proper K K perutato atrces A ad A such that ya Ay ad xa Ax. Sce A ad A are perutato atrces the followg holds: A A ' ad A A '. The vector correspodg to the {( )} ya A y ad lewse A Also A p p A p. x y p orderg ca be obtaed as A x A x ad A A p cota the ad the P eleets respectvely alged accordg to the {( )} x y p orderg. A If we wor out () (0) ad () by ag use of the property A 'A I we get C C C A A A μ N p ' ( A SA ' D ) p (9) p ' W S D p (0) ( ) W A A A A μ N p ' J W A SA ' D p () ( ) AG A A A A μ N where A W A W A ' ad ( ) ( ) D y 'A ' A y ' y ' y '. A A A W For what cocers C () t s show Appedx A that It s ot excluded that y > y g > h. h Here also t s ot excluded that x > x g > h. g h g

164 5 A. Verzz: Applyg the Hadaard WA ASA ' WA S. AG Focusg o C decoposto otce that: C C p ' J W S D p () ( ) B A A A A μ N p ' J W A SA ' S D p () ( ) ( ) T A A A A μ N Sug () ad () yelds (). W W AG AG B B T Should C G C G C G C G ad C (9) (0) () () ad () would tae the followg fors: G G G G ( ) A A A μ N G T the p ' A SA ' D p () p ' W S D p (5) ( ) W A A A A μ N p ' J W A SA ' D p (6) ( ) AG A A A A μ N G p ' J W S D p (7) ( ) B A A A A μ N p ' J W A SA ' S D p (8) ( ) ( ) T A A A A μ N DA y'a ' A y' y A' ya' ad WA A W A '. The atrx copact fors () (5) (6) (7) ad (8) correspod to the scalar expressos (9) (0) () () ad () respectvely. B We coclude ths secto by provdg closed-for expressos for C ad T AG C by bearg d C as specfed () uder the {( x y p ) } orderg: where ( ) ( ) C C p ' J W A 'SA D p (9) ( ) B μ N p ' J W S A 'SA D p (0) ( ) ( ) T μ N

165 STATISTICS IN TRANSITION-ew seres Deceber Redstrbuto ad re-rag dexes The redstrbutve effect of a tax syste ca be easured by the dfferece betwee the G dex for the pre-tax coe dstrbuto ad the G dex for the post- tax coe dstrbuto : followg e.g. Urba ad Labert (008) we shall deote dfferece by the acroy RE. The Atso-Plotc-Kawa dex s geerally appled to easure the rerag effect geerated by a tax syste; t s defed as the dfferece betwee the G dex for the post-tax coe dstrbuto ad the cocetrato dex for et coes the {( x y p ) } orderg. The Atso Plotc; Kawa dex s usually deoted by the acroy R. I cosderg the effects of a tax t ay be terestg to evaluate how RE ad R act wth ad across groups ad evetually also how they odfy both group average postos ad group tersectos. Ths ca be attaed by splttg ether RE or R to the wth groups across groups betwee groups ad trasvarato copoets troduced the prevous secto. Oe of the advatages of the copact expressos troduced the prevous sectos s that all dexes ca be calculated ether algg coes accordg to the pre-tax or accordg to the post-tax rag. We wll preset the RE ad the R x y p dexes by wrtg D atrces ad p vectors ether accordg to the {( )} or the {( )} x y p ordergs whe dvdual coe uts are cosdered. Here for the sae of shortess the decopostos of RE wll be reported oly x y p orderg ad coversely R decopostos wll accordg to the {( ) } be wrtte accordg to the {( ) } represeted ether accordg to the ( x y p ) or to the ( x y p ) ordergs... The RE dex Fro the defto of RE we ca wrte x y p orderg. All dexes could be also { } A { } A W AG W AG W B T W B T ( ) ( ) ( ) ( ) RE G G G G G G G G G G G G Rearragg ters we get ( ) ( ) RE G G G G RE RE W W AG AG W AG () See e.g. Labert (00 Ch. Secto.5). Plotc (98) Labert (00 Ch. Secto.5). The forulae that are ot reported ths artcle wll be provded to ayoe o request.

166 56 A. Verzz: Applyg the Hadaard Here what cocers AG RE bearg d that ( ) ( ) AG B T G G G we get AG B B T T B T RE G G G G RE RE () Fro (5) ad (6) t follows that RE G G ( ) ( ) μ μ p ' S D ESE' D p (a) N μ μ ( ) ( ) μ μ p' E'SE D S D p (b) N μ μ The RE W copoets ca be wrtte accordg to () ad bearg d W W W (0) as RE G G { ( ) ( ) } μ μ p ' W S D ESE' D p () N μ μ Lewse the RE AG copoets ca be wrtte as RE G G AG AG AG {( ) ( ) ( ) } μ μ p ' J W S D ESE' D p (5) N μ μ Resortg to (9) ad (0) B RE ad μ μ RE G G T RE ca be rewrtte as B B B {( ) ( ) N p ' J W μ A 'SA D ( ) } μ EA 'SAE' D p (6) RE G G T T T μ μ {( ) ( ) N p ' J W μ S A 'SA ( ) } D μe SA 'SA E' D p (7)

167 STATISTICS IN TRANSITION-ew seres Deceber The R (Atso-Plotc-Kawa) dex Fro the defto of R we ca wrte W AG W AG W B T W B T ( ) ( ) ( ) ( ) R G C G G C C G G G C C C Rearragg the ters we get ( ) ( ) R G C G C R R ad partcular for what cocers W W AG AG W AG (8) AG R we have ( ) ( ) AG B B T T B T R G C G C R R (9) Whe cosderg coe uts dvdually fro () () (5) ad (7) the dex R ad ts copoets ca be wrtte as follows R G C p' ( S E'SE ) D p (0a) μ N ( ) μ p ' ESE' S D p (0b) N Fro (0) ad (0a) t follows that ( ) R G C W W W μ p' W S E'SE D p () N Fro () ad (0a) t follows that ( ) ( ) R G C AG AG AG Fro (9) the copoet N μ p' J W S E'SE D p () N B R of R ca be expressed as R G C B B B ( ) ( ) μ p ' J W A 'SA E'A 'SA E D p () Fro (0) the copoet N T R ca be expressed as R G C T T T {( ) ( ) ( ) } μ p' J W S A'SA E'S A 'SA E D p ()

168 58 A. Verzz: Applyg the Hadaard AG B Ether fro the deftos of R ad R or by rearragg the ters () T R ca be gve the followg represetatos: ( ) p' {( J W ) T AG B R R R μ N ( ) ( ) } S E'SE A 'SA E'A 'SA E D p (5). The ssue of the sgs of R ad ts copoets We wll ow aalyse the sgs of R ad of ts decopostos by ag use of the atrx tools troduced the prevous sectos. Although ost of the results preseted ths secto are avalable the specalzed lterature we th that our reapprasal of the ssue through a talor-ade atrx toolt provdes soe addtoal sghts o the atter. Deostratos wll be carred out by spectg the quadratc for whch the R dex ad ts decopostos are proportoal to. R It s well ow that for the cocetrato C dex the property G C G holds fro whch t follows that R G C 0. Ths result wll be proved cosderg expresso (0a). Stateet The quadratc for p' ( S E'SE ) D p s o-egatve defte. Proof Recall that () atrx S s has all super-dagoal eleets equal to e ad sub-dagoal oes equal to ; () the eleets of E'SE s ay ot ecessarly respect the sae repartto as S due to perutatos perfored by E. Thus for all etres of S ad E'SE whch preset the sae values e e s s otherwse for < we would have s s ad for > 0 e s s. Bearg d that for < the atrx D d has superdagoal eleets o-egatve ad sub-dagoal oes o-postve the product Muss (008 Ch page 9) dscusses the sgs of R ad ts copoets R W R B ad R T. The author observes also that R T ca be postve ull or egatve the fraewor of o cotguous pre-tax coe groups: the proofs reported here coplete the author s stateets especally what cocers R T. See also Verzz (007) for cosderatos o G ad C copoets especally for pre-tax o overlappg groups. Kawa (980 Corollary 8.7 page 75).

169 STATISTICS IN TRANSITION-ew seres Deceber e ( ) s s d wll ay case result to be o-egatve whch proves the Stateet. R W ad R AG W W W AG AG AG We wll prove that R G C 0 ad R G C 0 by cosderg expressos () ad () respectvely. Stateet The quadratc fors p' W ( S E'SE ) D p ad p' ( ) ( ) J W S E'SE D p are o-egatve defte. Stateet s ust a corollary of Stateet. R B B B B We ow prove that R G C 0. I order to carry out the proof as for the prevous Stateets t s coveet to cosder a atrx copact for that correspods a straghtforward aer to the secod ter the rght had sde μ μ μ... μ ' of group averages of (5a). Let us defe the H vector [ H] (h.h) the H vector [ p p p ] μh μ h p h K h h p... ' of group weghts p ad the H H atrx D ( μ ' μ ' ) of group average dffereces. The G p's D p (6) H H B μ. h μ. g phpg μn h g μn where S s ow a H H atrx. After havg defed μ ad p respectvely as the H vector of μ h ad the H vector of atrx ( ) p h alged accordg to the {( )} H x y p order ad the H H A D μ ' μ ' () ca be rewrtte ths way: C p ' S D p (7) B μ N Fally by deotg by E the H H full ra perutato atrx such that μ Eμ μ E'μ p Ep ad p E'p R B ca be rewrtte as R G C p' S E'SE D p (8) ( ) B B B μ N

170 50 A. Verzz: Applyg the Hadaard Stateet The quadratc for p' ( S E'SE ) D p s.. defte. Proof Cosderatos aalogous to those reported above hold for ( S E'SE) D. I D the super-dagoal etres are o-egatve the sub-dagoal etres are o-postve: whle the forer are ultpled ether by 0 or by etres whch S E'SE the latter by 0 or by etres are the super-dagoal part of ( ) whch are the sub-dagoal part of ( S ) R B 0. E'SE ad hece t s proved that R T Dfferetly fro R R W G T ad R B that are all o-egatve R T ca be ether postve or egatve ad obvously equal to zero. Stateet 5 I expresso () the quadratc for {( ) ( ) ( ) } p' J W S A'SA E'S A 'SA E D p ca be zero postve or egatve. Proof Both atrx ( ) ( ) ω S A 'SA ω ad atrx S A 'SA o zero super-dagoal etres are o zero subdagoal are. Due to perutato perfored by E' ad E E' ( S A ) e 'SA E ω ca preset soe as super-dagoal etres ad syetrcally soe as sub-dagoal etres: hece ot cosderg the cases e whe both ω ad ω are zero the super-dagoal dffereces ( ) ( ) { e ω ω } S A 'SA E' S A 'SA E ay assue values [ ] [ ] 0 [ ] [ 0] [ ] [ ] [ 0] [ ] [ 0] [ ]. It follows that o-egatve super-dagoal etres of D ca be ultpled by a egatve value. Syetrcally sub-dagoal etres of ( SA 'SA) E' ( SA 'SA ) E ca ow be equal ot oly to [ ] [ 0] [ ] [ ] ad to [ 0] [ ] but also to [ 0] [ ] so that o-postve sub-dagoal etres of D ca be ultpled by a postve value whch proves the Stateet.

171 STATISTICS IN TRANSITION-ew seres Deceber Coclusos By use of the Hadaard product a elegat copact represetato atrx otato has bee obtaed ot oly for G cocetrato dexes ad for ther decopostos but for redstrbuto ad re-rag dexes ad ther decopostos as well. The atrx toolt troduced ths paper paves the way to obta foratve expressos for both the sad dexes ad ther copoets wth coes alged ether accordg to the pre-tax o-decreasg order or to the post-tax o-decreasg order. Moreover the copact represetato troduced ths paper leads to establsh a straghtforward aer the sgs of the Atso-Plotc-Kawa dex ad of ts copoets. We prove that R R W R AG ad R B are o-egatve quattes both whe pre-tax coe groups do overlap ad whe do ot. I the latter case R T G T (R T R AJL followg Urba ad Labert 008 otato) s o-egatve whereas the forer case we show R T ca be ether postve or egatve. Eve f t s well ow that R ad G T R AJL are o-egatve the proofs preseted ths paper are ew. Appedx O splfyg C W We wll prove the splfcato used forula (0) that s WA ASA ' WA S (A) Proof a The eleets w of atrx W ad the eleets w l ' l aw a of atrx W A are equal to f the assocated par of coes x ad x belog to the sae group they are zero otherwse. As all super-dagoal eleets atrx S are plus ad sub-dagoal eleets are we have to prove that all superdagoal eleets of atrx ASA ' that are selected by W A are ad all sub-dagoal eleets of ASA ' selected by W A are. Observe that coes belogg to the sae group rea raed a o x y p decreasg order wth each group also accordg to the {( )} orderg: therefore () f the ( x y p ) orderg x occupes the -th posto ad { } -th oe wth < the {( )} posto ad A x the -th oe wth l<; A x the x y p orderg x wll occupy the l-th

172 5 A. Verzz: Applyg the Hadaard () syetrcally the {( )} x y p orderg all pars of coes x > x belogg to the sae group wll respectvely be postos ad > x y p orderg postos l ad l> ad the {( )} A respectvely. a Ths ples that the etry s of S wll be shfted to the etry s l of ASA' wth l< f < ad l> f > so that the super-dagoal part of WA ASA ' all eleets wll be equal to ad the sub-dagoal part all eleets wll be equal to whch proves (A). REFERENCES ARONSON R. J. P. J. LAMBERT (99) Iequalty decoposto aalyss ad the G coeffcet revsted The Ecooc Joural 0 pp. 7. ARONSON R. J. P. J. LAMBERT (99) Decoposg the G coeffcet to reveal the vertcal horzotal ad re-rag effects of coe taxato Natoal Tax Joural 7 pp ARONSON R. J. P. J. JOHNSON P. J. LAMBERT (99) Redstrbutve effect ad uequal coe tax treatet The Ecooc Joural 0 pp BHATTACHARA N. B. MAHALANOBIS (967). Regoal dspartes household cosupto Ida Joural of the Aerca Statstcal Assocato 6 pp. 6. DAGUM C. (997) A ew approach to the decoposto of G coe equalty rato Eprcal Ecoocs pp FALIVA M. (98) Idetfcazoe e Sta el Modello Leare ad Equazo Sultaee Vta e Pesero Mlao. FALIVA M. (987) Ecooetra Prcp e Metod UTET Toro. FALIVA M. (996) Hadaard atrx product graph ad syste theores: otvatos ad role Ecooetrcs Matrces ad Graphs Theory ad Applcatos to Ecoocs S. Caz S. Stefa eds. World Scetfc Lodo. FALIVA M. (000) Su alcue fuzo geeralzzate utlzzate ell aals de process arozzabl Statstca 60 pp

173 STATISTICS IN TRANSITION-ew seres Deceber KAKWANI N. C. (980) Icoe Iequalty ad Poverty: Methods of Estato ad Polcy Applcatos Oxford Uversty Press. LAMBERT P. (00) The Dstrbuto ad Redstrbuto of Icoe Machester Uversty Press. LANDENNA G. (99) Fodaet d Statstca Descrttva l Mulo Bologa. MONTI M. A. SANTORO (007) The G decoposto: a alteratve forulato wth a applcato to tax refor ECINEQ d Coferece Berl avalable at ro.pdf. MONTI M. (007) O the Dagu decoposto of the G equalty dex DEAS Uverstà degl stud d Mlao W.P MONTI M. (008) A ote o the resdual ter R the decoposto of the G Idex Argueta Oecooca 0 pp MOOKHERJEE D. A. SHORROCK (98). A Decoposto Aalyss of the Tred UK Icoe Iequalty The Ecooc Joural 9 pp MUSSINI M. (008) La surazoe del rordaeto: aspett etodologc ed u applcazoe co rguardo a reddt delle fagle laes PhD thess Uverstà degl Stud d MIlao-Bcocca. PLOTNICK R. (98) A easure of horzotal equty Revew of ecoocs ad Statstcs PATT G. (976). O the Iterpretato ad desegregatos of G Coeffcet Ecooc Joural v RAO V. (969). The Decoposto of the Cocetrato rato Joural of the Royal Statstcal Socety 8 5. SCHOTT J. F. (005) Matrx Aalyss for Statstcs d edto Wley Hoboe New Jersey. SILBER J.(989). Factor CopoetsPopulato Subgroups ad the Coputato of the G Idex of Iequalty The Revew of Ecoocs ad Statstcs URBAN I. P. J. LAMBERT (008) Redstrbuto horzotal equty ad rerag: how to easure the properly Publc Face Revew 0. 0 pp.. VERNIZZI A. (007) Ua precsazoe sulla scoposzoe dell dce d redstrbuzoe RE d Aroso-Johso-Labert e ua proposta d estesoe

174 5 A. Verzz: Applyg the Hadaard dell dce d Plotc Ecooa Pubblca 7. pp. 5 5 ad DEAS Uverstà degl Stud d Mlao WP ITZAHAKI S. R. LERMAN (99). Icoe stratfcato ad coe equalty Revew of coe ad wealth 7 9. ITZAHAKI S. (99). Ecooc dstace ad overlappg of dstrbutos Joural of Ecooetrcs. 6 pp

175 STATISTICS IN TRANSITION-ew seres Deceber STATISTICS IN TRANSITION-ew seres Deceber 009 Vol. 0 No. pp BOOK REVIEW Tas force o the qualty of the labour force survey Fal report Eurostat ethodologes ad worg papers 009 edto 69 pages The Eurostat publcato o qualty of the labour force survey wll be useful to offcal statstcas ad acadec statstcas practtoers researchers aalytcs ad studet who are terested data qualty assesset ad proveet. The preset docuet reports the vews of the Tas Force o the Qualty of the Labour Force Survey (LFS) as a result of ts sx eetgs betwee Jue 007 ad Aprl 009. It also taes to accout the vews expressed by soe of the a Europea sttutoal users aely the Europea Cosso's Drectorate Geeral "Eployet Socal Affars ad Equal Opportutes" (DG EMPL) ad the Europea Cetral Ba (ECB) ad the feedbac fro the Labour Maret Statstcs (LAMAS) Worg Group Septeber 007 Aprl ad Septeber 008. The Tas Force was set up by the LAMAS Worg Group at ts March 007 eetg. It was coordated by Eurostat ad coposed of atoal delegates wth substatal expertse o the LFS fro e Meber States: Frace Geray Greece Italy the Netherlads Polad Portugal Spa ad the Uted Kgdo. Ths tatve s le wth the cotuous wor to prove the qualty of the LFS. It was coceved to cosoldate the gas acheved recet years ad reforce the status of the LFS due to ts hstory saple sze ad rchess of characterstcs as the a statstcal source o the labour aret. The goal The goal of the Tas Force was to revew the qualty of the LFS alog wth the desos of the qualty fraewor for statstcal output of the Europea Statstcal Syste (ESS) detect weaesses ad recoed proveets. The focus of the revew was o the estates of eployet ad ueployet as these are the ost relevat ad largely used dcators produced by the LFS. Followg ths revew the Tas Force forulated forty-three recoedatos o: saplg desg ad saplg errors

176 56 Tas force o the weghtg schees o-respose tervewers ad feldwor orgazato survey odes ad questoare forato for users coherece coparablty of eployet ad ueployet statstcs relevace of the ILO cocept of eployet ad ueployet teless ad puctualty. Saplg desg ad saplg errors As cocers saplg desg ad saplg errors the saple should be balaced over geographcal areas ad referece wees. Ths would both prove the atoal quarterly ad yearly estates ad crease the relevace of the LFS by eablg the producto of good othly estates. Target populato saplg frae ad populato estates Moreover target populato saplg frae ad populato estates should be cosstet ad up to date order to avod overcoverage ad udercoverage. The portace of harozed rotato patters whch allow coparable logtudal aalyss at Europea level was also hghlghted. Fally the eed was recogzed for a clarfcato of the wordg of the precso requreets Coucl Regulato 577/98 ad for a agreed ethod to assess coplace wth the Regulato. No-respose No-respose the EU EFTA ad caddate coutres s rather hgh (about 0% o average). It s usually selectve wth respect to eployet ad ueployet thus affectg the accuracy of ther estato. Recoedatos cover studyg prevetg ad correctg for o-respose. Iforato o the characterstcs of o-respodets should be regularly collected to assess ad adust for o-respose bas ad to prove feldwor strateges. Sutable tools to reassure respodets Sutable tools to reassure respodets (such as free-toll ubers or presetato letters) should be troduced wth a specal vew to crease the partcpato of o-atoals. The use of the wave approach ad of depedet tervewg should be cosdered to reduce respose burde. Weghtg schees Fally weghtg schees should tae to accout specfc characterstcs of o-respodets to correct for o-respose bas.

177 STATISTICS IN TRANSITION-ew seres Deceber Role of tervewers The role of tervewers s crucal for the accuracy of the survey results. Several recoedatos ad good practces cocerg tervewers' cotractual features trag otorg ad geeral o the feld-wor orgazato were detfed wth a vew to coo gudace as atoal arrageets cocerg these features ted to vary. I partcular order to boost otvato ad ze turover peraet professoal tervewers should be used ad ther reuerato should be adequate to ther crucal role for the qualty of the survey. Itervewers' trag should cover ot oly the survey cotet but also how to coduct the tervew ad to prevet o-respose. Perodc debrefg ad focus groups should be orgazed to revew ad tacle ssues. Itervews should be carred out as close as possble to the referece perod to avod recall probles ad support tely producto of results. Coputer-asssted questoares The LFS should always be carred out by coputer-asssted questoares gve that the tradtoal paper-tervewg ode s o loger sutable to cope wth the coplexty of the survey. However the pact of self-adstered electroc data collecto cludg web-based odes o the easureet of ILO labour status should be carefully vestgated. The use of xed odes should be cosdered the lght of possble gas relatg to respose rates burde ad costs alog wth lely the ode effects. I ay case ay chages to odes questoares ad other explaatory survey ateral should be carefully tested ad ther pact assessed before troducto. Lac of coherece Lac of coherece betwee LFS ad atoal accouts eployet estates s a aor cocer as t ay har the credblty of statstcs. I ths regard dstgushg betwee dffereces coverage scope ad deftos fro cossteces that ca be ascrbed to the accuracy of the dfferet statstcs s of the utost portace. For ths purpose the Tas Force recoeded the use of recoclato tables betwee LFS ad Natoal Accouts estates. The value of approprate coucato to users o the ature of coherece ad the eed to provde gudace o whch source fts whch purpose were also recogsed. Hgher put harozato The dea of ovg towards hgher put harozato s cosdered too dffcult for the oet because of atoal specfctes ad eeds. Coucl Regulato o. 577/998 together wth the prcples for the forulato of the questos o labour status lad dow Cosso Regulato 897/000 rea therefore the bass at Europea level for coparable statstcs o eployet ad ueployet. However the prcples should be revewed order to clarfy partcular abguous pots. Such clarfcatos should ot ecessarly ply chages the regulato (ecessarly va a ew legal act) but

178 58 Tas force o the should stead be provded as uch as possble worg docuets such as the explaatory otes. Itroducg ovatos Care should be tae whe troducg ovatos as these ca egatvely pact o coparablty of statstcs over te. Natoal statstcal sttutes should always adequately pla ad otor all chages tated ether by Eurostat or by coutres order to assess the statstcal effect o te seres. Cosstet te-seres should be produced ad dsseated at least for the headle dcators. For ts part Eurostat should group together ovatos t proposes order to lt the uber of potetal breas te seres. The relevace The relevace of the ILO labour force cocept was cofred although the eed for suppleetary dcators for the ILO ueployet rate both capturg a wder extet of the labour reserve ad allowg logtudal aalyss was recogzed. The varable "Ma Status as perceved by respodets" whch offers a copleetary vew to the ILO ecooc actvty status should be adatory the EU-LFS. The teless The teless of the EU-LFS ca be sgfcatly proved. Ths would further ehace ts relevace for short-ter ecooc aalyss. Establshg a release caledar would be slarly helpful. For ths purpose t s essetal that the twelve-wee deadle the Regulato as the oe for fal ot frst data trassso s respected. Recoeded practces All recoeded practces are effectve for provg the qualty of the LFS ad are feasble as they are already use at least oe coutry. Most of the recoedatos apply to atoal statstcal sttutes whereas several apply to Eurostat ad a few to both. The full lst of recoedatos grouped by subect s provded at the ed of the report. Page ubers bracets at the ed of each recoedato refer to the pot the text where they are dscussed. I hope that slar report wll be prepared for other saple surveys such the Household Budget Survey ad the EU-SILC (Statstcs of Icoe ad Lvg Codtos). Prepared by Ja Kordos Warsaw School of Ecoocs

179 STATISTICS IN TRANSITION-ew seres Deceber STATISTICS IN TRANSITION-ew seres Deceber 009 Vol. 0 No. pp REPORT The Deographc Future of Polad a scetfc coferece Łódź 7 8 Septeber 009 Betwee 7 ad 8 Noveber 009 the Char of Deography ad Socal Gerotology Uversty of Łódź orgazed Łódź a coferece ettled The Deographc Future of Polad. The coferece focused o future chages both deographc structures ad the processes that shape the. Chagg populato structures ad processes have short-ter as well as log-ter cosequeces. The creasg awareess of pacts duced by populato chages aes the latter ore terestg ot oly to deographers but also to researchers actve other felds of scece. Hece addto to deographers the 50 coferece partcpats represeted also socology socal polcy statstcs ad gerotology. The coferece was coceved to provde a platfor for presetg soe selected populato proectos ad exchagg opos o the pace drectos ad cosequeces of the aforeetoed chages. The research areas addressed the delvered papers ad durg the dscussos cocetrated aroud the below detaled topcs: Outcoes of the deographc proectos for Polad wored out by varous forecastg sttutos cludg the presetato of Polad agast the deographc ap of Europe. Ipacts of the chages populato structure affectg the labour aret socal securty systes health stuato. Chages the deographc structure wth respect to grato processes. The fluece of deographc chages o the evoluto of the cosuer goods ad servces aret ad predctg future tedeces ths area. The ethods for forecastg deographc processes. The coferece was opeed by Prof. Jolata Grotowsa-Leder Assocate Dea at the Faculty of Ecoocs ad Socology Uversty of Łódź Prof. Czesław Doańs Drector of the Isttute of Statstcs ad Deography ad Prof. Agesza Rossa Head of the Char of Deography ad Socal Gerotology. The coferece cossted of fve theatc sessos. The frst sesso The Future of the Faly ad Procreatve Behavour was chared by Prof. Zofa Zarzyca. The dscussed topcs cocered ot oly the chages procreatve

180 50 The Deographc Future behavour ad ther effect o the future codto ad structure of populato but also faly polcy ad ts challeges. The relevat cosequeces were preseted by Dr. Mlea Lage the cotext of the results of deographc forecasts. The presetato delvered by Prof. Adrze Ochoc ad Marta Kawńsa M.Sc. dealt wth faly polcy probles a broader Europea cotext. Dr. Potr Szuals s delberatos cocetrated o factors that could eable the Polsh populato to retur to the strct replaceet rate. I hs opo the fertlty rate wll ot exceed.7 the ear future although as the author stressed a slfully pursued proatalst polcy could help reverse ths ufavourable tred the ext decades. Aother paper preseted durg the sesso cocered the fluece of relgousess o the faly plag decsos. Aa Madzńsa M.Sc. ad Wtold Śgels M.Sc. coducted a questoare survey aog studets of the th ad 5th grades of the Uversty of Łódź to explore how they evsage ther faly future. The leadg thee of the secod sesso chared by Prof. Adrze Ochoc was the fluece of chagg structure ad age of populato o the labour aret. Most speaers durg the sesso ettled The Deographc Deterats of Chagg Stuato of Households ad Labour Resources focused o the proble of agg labour force ad ts cosequeces for both eployers ad eployees. Oe aspect addressed durg ths part of the coferece was the ecooc actvty of Poles the pre-retreet perod whch s the lowest the EU. It was dscussed by Prof. Hala Worach-Kardas ad Szyo Kostrzews M.Sc. The authors beleve that the hgh rate of ueployet aog persos aged 50 s caused by obectve factors such as worers lac of relevat occupatoal slls ad health codto ad subectve deterats cludg preudce agast older worers. Katarzya Baładyowcz-Pafl M.Sc. was of the opo that age aageet could be a respose to the agg of labour resources. Ths ter ecapsulates actos aed at provg older persos stuato the worplace e.g. volvg the troducto of flexble worg te trag ad worplace adaptato to the eeds of older worers. It should be reebered though as stressed by the author that these actos ust be tae ot oly by the eployers but also by the goveretal sttutos. Eployet flexblty that Dr. Zofa Szyae referred to s oe of the ost portat struets capable of prevetg further reducto the actvty of persos at pre-retreet age. A creased proporto of flexble wor offers s lely to help actvate persos who have preaturely wthdraw fro the labour aret. The thrd sesso The Trasforg Dead for Health ad Socal Beefts was chared by Prof. Ireeusz Kuropa. Ths was aother sesso where the agg of Polad s populato ad the process cosequeces for healthcare ad log-ter ad were the doat thees. Health codto ad orbdty aog persos aged 65 years ad older were addressed the presetato prepared by Olga Gaewsa M.Sc. Prof. Irea Maeca-Bryła ad Dr. Mare Bryła. Ther aalyss provded a sght to the stuato of persos

181 STATISTICS IN TRANSITION-ew seres Deceber lvg the couty of Płoc ad usg the servces of the prary health care physcas. The other speaers the sesso cocetrated o gvg care to seors. Factors deterg the tae-up of socal ad servces by persos aged 60 years ad older were preseted ad aalysed by Moa Szlawsa M.Sc. Prof. Irea Maeca-Bryła ad Dr. Mare Bryła. The authors studed the populato lvg the tow ad the rural coue of Zgerz. The ext paper dealt wth facg care servces for the elderly ad vestgated the log-ter care surace a subect that s frequetly rased durg dscussos o old age. Dr. Barbara Debowsa poted out that the growg populato of older persos partcularly of seors the fourth age would cosequetly crease the dead for care that the -faly support wll ot be able to eet. The growg uber of persos eed of assstace wll etal the proble of eararg larger fuds for ad servces exteded to older persos. Zofa Szweda-Lewadowsa M.Sc. attepted her tal to estate the future dead for beds ursg hoes aog persos aged 75 years ad older. The secod day of the coferece was broe dow to two sessos. The frst of the The Soco-Ecooc Challeges a Agg Socety was chared by Dr. Ala Potryowsa. The sesso cotued to explore the agg of Polad s populato that was already dscussed o the frst day of the coferece. The presetato opeg the sesso was delvered by Prof. Barbara Szatur-Jaworsa ad ts subect was socal polcy towards old people exeplfed by the case of Warsaw. The author ephaszed that oe of the ey cocers should be a correct dagoss of the stuato of seor ctzes oe provdg a startg pot for forulatg prograes for the subpopulato of old persos. The stuato of old persos ad the qualty of ther lves were also evaluated by Dr. Dorota Jachowcz-Wołoszye. The speaer had studed soe attrbutes of seors such as ther geder age ad educatoal attaet the cotext of ther fluece o the assesset of lfe qualty. Further Prof. Berard Rzeczyńs troduced ssues related to gerototechology ad dscussed how urba layout terferes wth the fuctog of the elderly urba space. Dr. Dorota Kozeł ad Dr. Małgorzata Kaczarczy preseted the challeges that ageg socety faces the area of educato ad prevetg the excluso of seors. Accordg to the authors a strategy prootg lfe-log learg eeds to be pleeted so that the soco-ecooc actvty of persos older age groups ca be supported. Karola Jasólsa M.Sc. was aother speaer cosderg the stuato ad role of old persos socety. She uderled that the ear future the role of seors a fast-chagg owledge-based socety wll appear ore ad ore ofte socal debates. Aother presetato delvered by Prof. Agesza Rossa dealt wth ortalty forecastg. The author proposed to use the dyac lfe tables based o the Lee-Carter odel coputg the aouts of pesos to be draw by future old-age pesoers. The last presetato delvered durg the sesso dscussed the role of the aret for cosuer goods ad servces used by the elderly. Patryca Woszczy M.Sc. dscussed both the preset stuato the area ad the relevat forecasts.

182 5 The Deographc Future The closg sesso of the coferece chared by Prof. Jerzy Kowales cocered the teratoal deso of deographc chages Polad. I ther presetato Prof. Eugeusz Zdroews ad Małgorzata Guzńsa M.Sc. dscussed the sze ad destatos of peraet grato Polad. Aog other thgs the authors poted to the proble of egrats wth hgh occupatoal qualfcatos. The cetral thee of the tal preseted by Dr. Ala Potryowsa was the future developet of deographc processes Polad the cotext of grato. Prof. Jerzy Kowales the charperso of the last sesso cocluded the coferece by stressg the portace of the topcs addressed ad potg to the dversty of ssues that were rased the presetatos. Although ageg ad ts cosequeces doated aogst the topcs fact all the a areas of terest to deographers were preseted. Speag o behalf of the coferece orgazers Prof. Agesza Rossa thaed the partcpats for ther attedace ther presetatos ad actve partcpato the dscussos. She also vted the to tae part aother coferece that the Char of Deography ad Socal Gerotology UŁ s orgazg Łódź Lfe Qualty of the Elderly the Presece ad the Future (- Jue 00). All forato relevat to the forthcog coferece ca be sought o the webste: ad Zofa Szweda-Lewadowsa Agesza Rossa

183 STATISTICS IN TRANSITION-ew seres Deceber STATISTICS IN TRANSITION-ew seres Deceber 009 Vol. 0 No. pp REPORT VIII Coferece o Multvarate Statstcal Aalyss MSA 009 Łódź Polad 6 8 Noveber 009 The VIII Coferece o Multvarate Statstcal Aalyss was held fro 6th to 8th Noveber 009 Łódź. Orgazato of the coferece was charged to the Char of Statstcal Methods Uversty of Łódź ad Polsh Statstcal Assocato. The coferece preseted the latest theoretcal acheveets the feld of the ultvarate statstcal aalyss ad ts applcatos. Ths s a cotuato of the ssues udertae o the cofereces orgazed the past years. The scetfc prograe of MSA 009 covered a rage of statstcal probles such as: ultvarate dstrbutos statstcal tests oparaetrc ferece dscrato aalyss Mote Carlo aalyss Bayesa ferece applcato of statstcal ethods face surace captal arets ad rs aageet. Ths year the ope eetg of Cottee of Statstcs ad Ecooetrcs PAN was held as part of the Coferece. O the eetg Prof. Agesza Rossa delvered a lecture Stochastc odels of populato dyacs. Altogether there were 70 partcpats fro varous acadec ad research cetres Polad. Cocerg the papers papers were preseted sessos. The cofereces were opeed by the Chara of the Orgazg Cottee: Prof. Czesław Doańs. The opeg speech was also gve by the Pro-vce Chacellor of the Uversty of Łódź Prof. Ato Różals ad the Dea of the Faculty of Ecoocs ad Socology of the Uversty of Łódź Prof. Ja Gada. The frst pleary sesso (char: Prof. Jausz Wywał) was devoted to faous Polsh statstcas: Stasław Marc Ula (909 98) ad Ludw Krzywc (859 9). Prof. Mrosław Krzyśo (Ada Mcewcz Uversty Pozań) reported a paper ttled Staslaw Marc Ula. Staslaw Marc Ula (bor Aprl 909 Lwów ded May 98 Sata Fe New Mexco U.S.) was a Polsh ad Aerca Matheatca ( 9 he becae a U.S. ctze) who helped to develop the Teller-Ula desg whch powers the hydroge bob as well as a uber of other portat atheatcal tools (brachg processes Mote Carlo ethod). Prof. Czesław Doańs (Uversty of Łódź) delvered a lecture about Ludw Krzywc. Ludw Krzywc (bor August 859 Płoc ded

184 5 VIII Coferece o Multvarate Jue 0 9 Warsaw) was a Polsh athropologst ecoost ad socologst. Oe of the early chapos of socology Polad he approached hstorcal aterals fro socologcal vewpot. Krzywc studed atheatcs at the Uversty of Warsaw Polad. Later he bega studyg athropology archaeology ad ethology aog others Pars. Fro 99 to 96 he was a professor at the Uversty of Warsaw. The ttles of the papers of the ext sessos of the coferece MSA wth the aes of the authors are preseted below: 6 Noveber 009: Pleary Sesso II: Char: Prof. Mrosław Krzyśo Sulato aalyss of accuracy estato of populato ea o strategy depedet o order statstc of auxlary varable (Jausz L. Wywał Katowce); Coparso betwee prcpal copoet aalyss ad factor aalyss: a foratoal perspectve (Therry Dhore Fraca- Vaes ); Aalytcal terpretato of orespose error (Mrosław Szreder Gdańs). Sesso III A: Char: Prof. Mrosław Szreder The coparso of odel based clusterg wth the heurstc clusterg ethods (Ewa Wte Katowce); The fluece of rrelevat varables o classfcato error rules ducto (Marusz Kubus Opole); Coparso of stablty of algorths classcal ad eseble approach taxooy (Dorota Rozus Katowce); Shapley value regresso for oparaetrc ultple putato (Ewa Nowaowsa Krzysztof Puszcza GFK Poloa); Multple classfcato aalyss deography (Alesader Suseł Nowy Sącz). Sesso III B: Char: Prof. Wesław Wager Trasforato of ecooc statstcs (Elżbeta Gołata Grażya Dehel Pozań); Deterats of supply chas of blood Polad (Przeysław Jezors Sebasta Twaróg Katowce); Iovatos ad usage of ew techologes Polsh sall ad eduszed eterprses results of survey (Toasz Jurewcz Daa Gada Gdańs);

185 STATISTICS IN TRANSITION-ew seres Deceber The use of ba servces by sall ad edu-szed eterprses before ad after the facal crss (Toasz Jurewcz Daa Gada Gdańs); The estato of the corrupto percepto dex (Alesadra Baszczyńsa Dorota Peasewcz Łódź). 7 Noveber 009: Pleary Sesso I: Char: Prof. Jausz Wywał Soe tests for quatle regresso odels (Grażya Trzpot Katowce); Coparso of estators of a probablty of success two odels (Wocech Zelńs Warszawa); O effectveess of Hellwg's varable choce ethod lear regresso odel (Tadeusz Bedars Flp Borowcz Wrocław). Sesso II A: Char: Prof. Grażya Trzpot Model selecto crtera for reduced ra ultvarate te seres wth applcato detfcato of perodc copoets (Marc Hława Mace Kawec Wrocław) Depth based strateges to robust estato of ARIMA paraeters (Dael Kosorows Kraów); Coparso of selected ethods for varable selecto support vector aches (Mchał Trzęso Katowce). Sesso II B: Char: Prof. Agesza Rossa Coparso of selected ethods for varable selecto support vector aches (Wesław Wager Adrze Mata Rzeszów); O soe ssues the practce of doa fracto predcto (Toasz Żądło Katowce); Sustaable developet regoal deso soft odel (Dorota Perło Bałysto). Sesso III A: Char: Prof. Wocech Zelńs Multdesoal soothg tables of fertlty rates (Toasz Jurewcz Gdańs); Two-saple tests for crossg survval curves at sall saples (Toasz Jurewcz Ewa Wyca Gdańs); Socoecooc well-beg soft odel (Dorota Merzyńsa Bałysto);

186 56 VIII Coferece o Multvarate A aalyss of ob seorty aog ueployed. Applcato of odel (Beata Jacowsa Ewa Wyca Gdańs). Sesso III B: Char: Prof. Krystya Katulsa Optu checal balace weghg desg uder certa codto (Brosław Ceraa Małgorzata Graczy Pozań); Note o the optu checal balace weghg desg for pv obects (Brosław Ceraa Małgorzata Graczy Pozań); Characterstcs of two-desoal boal dstrbuto (Wesław Wager Rzeszów); O the otorg of the process ea based o the sequece of perutato tests (Grzegorz Kończa Katowce). 8 Noveber 009: Pleary Sesso I Char: Prof. Mrosław Krzyśo Testg for tal depedece extree value odels applcato o Polsh stoc exchage (Grażya Trzpot Justya Maewsa Katowce); Extree value dex of left ad rght tals for facal te seres (Wesław Dzubdzela Mchał Stachura Barbara Wodeca Kelce); Choosg varables cluster aalyss by eas of etropy ad detectg uodal dstrbutos (Jerzy Korzeews Łódź). Sesso II A: Char: Prof. Krystya Prusa The fluece of the saple sze o the et preu rates car lablty surace CR (Aa Szyańsa Łódź); Costructo of the aggregatve dex of wor effcecy (Jace Bałe Łódź); Statstcal aalyss of the effcecy of peso systes of the EU ad EFTA (Artur Mulec Łódź). Sesso II B: Char: Prof. Krystya Katulsa Coparso of ethods deterato loss dstrbuto fucto fro credt portfolo CredtRs odel (Agesza Petrza Łódź); No-saplg errors opo polls (Alesadra Fałowsa Łódź).

187 STATISTICS IN TRANSITION-ew seres Deceber Pleary Sesso III: Char: Prof. Brosław Ceraa The role of o-facal eterprses Polad (Czesław Doańs Magdalea Motyl Łódź); The estato of the Steltes trasfor of spectral fuctos of covarace atrx (Aa Wtaszczy Łódź); The role of probablty theory ad statstcs ethodology of the exact sceces (Jausz Kupczu Łódź). The ext Coferece o Multvarate Statstcal Aalyss wll be held o Noveber Lodz. Scetsts terested attedg the Coferece are dly requested to sed ther applcato to the Scetfc Secretary of MSA 00 to the followg address: Aa Wtaszczy 9 th Coferece MSA 00 Char of Statstcal Methods Uversty of Łódź 90- Łódź Rewoluc 905 r. r Polad phoe: (8) ; fax: (8) e-al: [email protected] Katarzya Boloe-Lasoń Moa Zelńsa-Stewcz

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