An International Journal of the Polish Statistical Association

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1 STATISTICS IN TRANSITION nw sris An Intrnational Journal of th Polish Statistical Association CONTENTS From th Editor... Submission information for authors... 5 Sampling mthods and stimation CIEPIELA P., GNIADO M., WESOŁOWSKI J., WOJTYŚ M., Dynamic K-Composit stimator for an arbitrary rotation schm... 7 SHUKLA D., PATHAK SH., THAKUR N.S., Estimation of population man using two auxiliary sourcs in sampl survys... TAILOR R., SHARMA B., Modifid stimators of population varianc in prsnc of auxiliary information TIKKIWAL G.C., Khandlwal A., Crop acrag and crop production stimats for small domains rvisitd ONYEKA A.C., Estimation of population man in post-stratifid sampling using known valu of som population paramtr(s Othr articls CASTELLANOS E. M., Nonrspons bias in th Survy of Youth Undrstanding of Scinc and Tchnology in Bogotá SANKLE R., SINGH J.R., MANGAL I.K., Cumulativ sum control charts for truncatd normal distribution undr masurmnt rror GOŁATA E., Data intgration and small domain stimation in Poland xprincs and problms Comparativ Survys TARKA P., Customrs rsarch and quivalnc masurmnt in factor analysis Congrss of Polish Statistics: Th 00th Annivrsary of th Polish Statistical Association Editor s not on th Statistical Congrss Sction CZEKANOWSKI J. ( Biographical not... 6 NEYMAN J. ( Biographical not DOMAŃSKI CZ., 00 yars of th Polish Statistical Association ŁAGODZIŃSKI W., Th Polish Statistical Association (PTS R-stablishing SZREDER M., Nw conomy nw challngs for statistics... 9 KORDOS J., "Statistics in Transition" and "Statistics in Transition - nw sris" - First Fiftn Yars Editorial Offic. Statistics in Transition nw sris today... 0 Congrss Information Congrss Announcmnt Congrss Agnda Volum 3, Numbr, March 0 SPECIAL ISSUE

2 STATISTICS IN TRANSITION-nw sris, March 0 STATISTICS IN TRANSITION-nw sris, March 0 Vol. 3, No., pp. 4 FROM THE EDITOR Th Spring 0 issus of th Statistics in Transition nw sris is bing rlasd somwhat arlir than usually in ordr to contribut in this way to th upcoming Congrss of Polish Statistics, which is undr prparation to clbrat th hundrdth annivrsary of stablishing of th Polish Statistical Association. Accordingly, in addition to Journal s rgular sctions on stimation and sampling issus and othr articls, and also on comparativ survys a spcial congrssional sction is includd in this volum, containing voics ( occasional statmnts of svral mmbrs of th Journal s Editorial Board, and of important Congrss information matrials. Th first part starts with papr by Przmysław Cipila, Małgorzata Gniado, Jack Wsołowski and Małgorzata Wojtyś on Dynamic K-composit stimator for an arbitrary rotation schm. Authors bgin with an ovrviw of th proprtis of classical K-composit stimator proposd by Hansn, Hurwitz, Nisslson and Stinbrg (955 and intnsivly studid in Rao and Graham (964. It givs an altrnativ solution to quasi-optimal stimation undr rotation sampling whn it is allowd that units lav th sampl for svral occasions and thn com back. Sinc th K-composit stimator suffrs from crtain disadvantags as bing dsignd for a stabl situation in th sns that its basic paramtr is kpt constant on all occasions and rstrictd only to a crtain family of rotation dsigns authors propos a dynamic vrsion of th K-composit stimator (DK-composit stimator, without any rstrictions on th rotation pattrn. Although th proposd algorithm is simplr than th on for th classical K-composit stimator with optimal wights, it is prcis, in th sns that it dos not us any approximat or asymptotic approach. Diwakar Shukla, Sharad Pathak and Narndra Singh Thakur in papr ntitld Estimation of Population Man Using Two Auxiliary Sourcs in Sampl Survys propos familis for stimation of population man of th main variabl using th information on two diffrnt auxiliary variabls, undr simpl random sampling without rplacmnt (SRSWOR schm. Thr diffrnt classs of stimators ar constructd and xamind with a compltiv study with othr xisting stimators. Th xprssion for bias and man squard rror of th proposd familis ar obtaind up to first ordr of approximation. Usual ratio

3 W.Okrasa: From th Editor... stimator, product stimator, dual to ratio stimator, ratio-cum-product typ stimator and many mor stimators ar idntifid as particular cass of th suggstd family; thortical rsults ar supportd by numrical xampls. In th nxt papr, Modifid Estimators of Population Varianc in Prsnc of Auxiliary Information by Rajsh Tailor and Balkishan Sharma proposd is an stimator of population varianc using information on known paramtrs of auxiliary variabl. It has bn shown that using modifid sampling fraction th proposd stimators ar mor fficint than th usual unbiasd stimator of population varianc and usual ratio stimator for population varianc undr crtain givn conditions. Empirical study is also carrid out to dmonstrat th mrits of th proposd stimators of population varianc ovr othr stimators considrd in this papr. G. C. Tikkiwal and Alka Khandlwal in papr Crop Acrag and Crop Production Estimats for Small Domains Rvisitd discuss th problm of advanc and final stimats of yild of principal crops, at national and rgional (Stat lvls, which ar of grat importanc for country s macro lvl planning. For dcntralizd planning and for othr purposs lik crop insuranc, loan to farmrs, tc., th rliabl stimats of crop production for small domains ar also in grat dmand. This papr, thrfor, discusss and rviw critically th mthodology usd to provid crop acrag and crop production stimats for small domains, basd on indirct mthods of stimation, including th SICURE modl approach. Th indirct mthods of stimation so dvlopd us data obtaind ithr through traditional survys, lik Gnral Crop Estimation Survys (GCES data, or a combination of th survys and satllit data. In papr Estimation of Population Man in Post-Stratifid Sampling Using Known Valu of Som Population Paramtr(s by A.C. Onyka a gnral family of combind stimators of th population man in post-stratifid sampling (PSS schm is prsntd, following Khoshnvisan t.al. (007 and Koyuncu and Kadilar (009, and using known valus of som population paramtrs of an auxiliary variabl. Proprtis of th proposd family of stimators, including conditions for optimal fficincy, ar obtaind up to first ordr approximations, and th rsults ar illustratd mpirically. Th scond group of articls ( othr articls is opnd by papr of Edgar Mauricio Buno Castllanos on Nonrspons Bias in Th Survy of Youth Undrstanding of Scinc and Tchnology in Bogotá. Th Colombian Obsrvatory of Scinc and Tchnology OCyT dvlopd in 009 a survy about undrstanding of Scinc and Tchnology in studnts of high school in

4 STATISTICS IN TRANSITION-nw sris, March 0 3 Bogotá, Colombia. Th sampling dsign was stratifid according to th natur of school (public or privat. Two sourcs of unit nonrspons wr dtctd. Th first on corrsponds to schools that did not allowd to collct information. Th scond sourc corrsponds to studnts who did not assist during th days whn survy was applid. Estimats wr obtaind through two diffrnt approachs. Rsults obtaind in both cass do not show visibl diffrncs whn stimating ratios; vn though, som grat diffrncs wr obsrvd whn stimating totals. Rsults obtaind using th scond approach ar blivd to b mor rliabl bcaus of th mthodology usd to handl itm nonrspons. R. Sankl, J.R. Singh and I.K. Mangal in papr Cumulativ Sum Control Charts For Truncatd Normal Distribution undr Masurmnt Error constructd Cumulativ Sum (CUSUM Control Charts for man undr truncatd normal distribution and masurmnt rror. For diffrnt truncation points and diffrnt sizs of masurmnt rror tabls hav bn prpard for th avrag run lngth, lad distanc and th angl of mask. Thy analyz th snsitivity of th paramtrs of th V-Mask and th Avrag Run Lngth (ARL through numrical valuation for diffrnt valus of r. Elżbita Gołata s papr Data Intgration and Small Domain Estimation in Poland Exprincs and Problms has twofold objctiv, ncompassing, on th on hand, a prsntation of Polish xprincs with th mthodological issus considrd currntly as on of th most important i.., data intgration (DI and statistical stimation for small domains (SDE; and, on th othr hand, it attmpts to dtrmin rlationship btwn ths two typs of mthods. Givn convrgnc of th goals of both mthods, SDE and DI (i.., to incras fficincy of th us of xisting sourcs of information, simulation study was conductd in ordr to vrify th hypothsis of synrgis rfrring to combind application of both groups of mthods: SDE and DI. Th third sction, comparativ survys, is rprsntd in this volum by on itm, by Piotr Tarka s papr on Customrs Rsarch and Equivalnc Masurmnt in Factor Analysis. Author discusss th problm of btwn population validity of th masurmnt, whn xtractd factors may hard to b qually compard on th rflctiv basic lvl (unlss all conditions of invarianc masurmnt ar mt. Hnc, implmntation of customrs rsarch and any intr-cultural studis rquir a multi-cultural modl dscribing statistical diffrncs in both culturs with invarianc as undrlying assumption. In th articl mployd was a modl for analysis of customrs prsonal valus prtaining to hdonic consumption aspcts in two culturally opposit populations.

5 4 W.Okrasa: From th Editor... Data wr gnratd through survy conductd in two countris, in th following citis: Poland (Poznan and Th Nthrlands (Rottrdam and Tilburg, using probability sampls of youth. This modl mad it possibl to tst invarianc masurmnt undr cross-group constraints and thus xamining structural quivalnc of latnt variabls valus. Th sction dvotd to th Congrss of Polish Statistics, concluds this volum. Włodzimirz OKRASA Editor-in-Chif

6 STATISTICS IN TRANSITION-nw sris, March 0 5 STATISTICS IN TRANSITION-nw sris, March 0 Vol. 3, No., pp. 5 SUBMISSION INFORMATION FOR AUTHORS Statistics in Transition nw sris (SiT is an intrnational journal publishd jointly by th Polish Statistical Association (PTS and th Cntral Statistical Offic of Poland, on a quartrly basis (during it was issud twic and sinc 006 thr tims a yar. Also, it has xtndd its scop of intrst byond its originally primary focus on statistical issus prtinnt to transition from cntrally plannd to a markt-orintd conomy through mbracing qustions rlatd to systmic transformations of and within th national statistical systms, world-wid. Th SiT-ns sks contributors that addrss th full rang of problms involvd in data production, data dissmination and utilization, providing intrnational community of statisticians and usrs including rsarchrs, tachrs, policy makrs and th gnral public with a platform for xchang of idas and for sharing bst practics in all aras of th dvlopmnt of statistics. Accordingly, articls daling with any topics of statistics and its advancmnt as ithr a scintific domain (nw rsarch and data analysis mthods or as a domain of informational infrastructur of th conomy, socity and th stat ar appropriat for Statistics in Transition nw sris. Dmonstration of th rol playd by statistical rsarch and data in conomic growth and social progrss (both locally and globally, including bttr-informd dcisions and gratr participation of citizns, ar of particular intrst. Each papr submittd by prospctiv authors ar pr rviwd by intrnationally rcognizd xprts, who ar guidd in thir dcisions about th publication by critria of originality and ovrall quality, including its contnt and form, and of potntial intrst to radrs (sp. profssionals. Manuscript should b submittd lctronically to th Editor: sit@stat.gov.pl., followd by a hard copy addrssd to Prof. Wlodzimirz Okrasa, GUS / Cntral Statistical Offic Al. Nipodlgłości 08, R. 87, Warsaw, Poland It is assumd, that th submittd manuscript has not bn publishd prviously and that it is not undr rviw lswhr. It should includ an abstract (of not mor than 600 charactrs, including spacs. Inquiris concrning th submittd manuscript, its currnt status tc., should b dirctd to th Editor by mail, addrss abov, or w.okrasa@stat.gov.pl. For othr aspcts of ditorial policis and procdurs s th SiT Guidlins on its Wb sit:

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8 STATISTICS IN TRANSITION nw sris, March 0 7 STATISTICS IN TRANSITION nw sris, March 0 Vol. 3, No., pp. 7-0 DYNAMIC K-COMPOSITE ESTIMATOR FOR AN ARBITRARY ROTATION SCHEME Przmysław Cipila, Małgorzata Gniado, Jack Wsołowski 3 and Małgorzata Wojtyś 4 ABSTRACT Classical K-composit stimator was proposd in Hansn t al. (955. Its optimality proprtis wr dvlopd in Rao and Graham (964. This stimator givs an altrnativ solution to quasi-optimal stimation undr rotation sampling whn it is allowd that units lav th sampl for svral occasions and thn com back. Such situations happn frquntly in ral survys and ar not covrd by th rcursiv optimal stimator introducd by Pattrson (955. Howvr th K-composit stimator suffrs from crtain disadvantags. It is dsignd for a stabl situation in th sns that its basic paramtr is kpt constant on all occasions. Additionally it is rstrictd only to a crtain family of rotation dsigns. Hr w propos a dynamic vrsion of th K-composit stimator (DK-composit stimator without any rstrictions on th rotation pattrn. Mathmatically, th algorithm, w dvlop, is much simplr than th on for th classical K-composit stimator with optimal wights. Morovr, it is prcis, in th sns that it dos not us any approximat or asymptotic approach (opposd to th mthod usd in Rao and Graham (964 for computing optimal wights.. Introduction It is wll known that, whil looking for optimal stimators in survys which rpat in tim with th sam tim spacing, taking undr account obsrvations not only from th prsnt dition of th survy (occasion but also from prvious occasions may significantly improv th quality of stimation. Bank PEKAO, Warszawa, POLAND Towarzystwo Ubzpiczń na Życi "Warta", Warszawa, POLAND 3 Główny Urząd Statystyczny and Wydział Matmatyki i Nauk Informacyjnych, Politchnika Warszawska, Warszawa, POLAND, -mail: wsolo@mini.pw.du.pl 4 Wydział Matmatyki i Nauk Informacyjnych, Politchnika Warszawska, Warszawa, POLAND, -mail: wojtys@mini.pw.du.pl

9 8 P. Cipila, M. Gniado, J. Wsołowski, M. Wojtyś: Dynamic K-composit... For th bst linar unbiasd stimators (BLUEs of th man on a givn occasion, to rduc tim and mmory rquirmnts, it is dsirabl to hav a rcursiv form for such an stimator which rfrs only to crtain (possibly, small numbr of optimal stimators from rcnt occasions and, additionally, obsrvations from thos occasions. Such a problm was compltly solvd in a sminal papr by Pattrson (950 for a family of rotation pattrns which do not allow for a com-back of a unit to th sampl, aftr laving it for som occasions. Th solution givs a formula for th BLUE of μ h, th man on th hth occasion, as a linar combination of th BLUE of μ h and obsrvations from (h th and hth occasions. Howvr, in many practical survys, th rotation pattrn allows hols, i.. som units stay in a sampl for a numbr of occasions, lav it for a numbr of occasions, thn rturn to th survy for a numbr of occasions. Important xampls includ th Currnt Population Survy (CPS in th US, whr th units follow th pattrn (a unit is in th sampl for subsqunt 4 occasions, lavs it for subsqunt 8 occasions, is again in th sampl for subsqunt 4 occasions and thn nvr rturns to th sampl, or polish Labour Forc Survy with th pattrn 00 (s Szarkowski and Witkowski (994 or Popiński (006. Unfortunatly th rcurrnt form of th BLUE in such situations of rotation pattrns with hols is not known in gnral, s for instanc Yansanh and Fullr (998. (Actually, th rcurrnt form of th optimal stimators for any rotation pattrn with hols of siz has bn drivd only rcntly in Kowalski (009 and for th Szarkowski schm 00 vn mor rcntly in Wsołowski (00. A widly accptd solution in th gnral situation is th K-composit stimator introducd in Hansn t al. (955. Its optimality proprtis wr studid for svral modls in Rao and Graham (964 (shortnd to RG in th rst of this papr. By dfinition K-composit stimator maks us only of th most rcnt past composit stimator and obsrvations from th prsnt and th most rcnt past occasions. Mor prcisly K-composit stimator on hth occasion, ˆμ h, has th following form ( ˆμ h = Q ˆμ h X (h h,h (h X h,h ( Q X h, ( whr ˆμ h is th K-composit stimator on (h th occasion, X(h h,h is th sampl man for th units common to both (h th and hth occasions calculatd for th hth occasion, X(h h,h is th sampl man of for th units common to both (h th and hth occasions calculatd for th (h th occasion, Xh is th sampl man for all th units on hth occasion and Q [0, is a numrical paramtr which dos not dpnd on h(!. Additionally in RG only a rstrictd though natural family of rotation pattrns is invstigatd:

10 STATISTICS IN TRANSITION nw sris, March 0 9 a group of units rmains in th sampl for r occasions, thn lavs it for m occasions, coms back to th sampl for r occasions, lavs it for m occasions, and so on. In such a stting strngthnd by assuming xponntial (Modl or arithmtic (Modl corrlation pattrn th optimal choic of Q is considrd in that papr (in passing, lt us not that Modl 3 for corrlation pattrn is impossibl sinc th rsulting covarianc matrix may not b positiv dfinit. To attain this goal in RG it is takn h, sinc othrwis, apparntly, th optimal Q has to dpnd on h. Numrical solutions ar thn obtaind sinc th rsulting formula ((4 in RG for th varianc of th stimator is analytically non-tratabl. As it alrady has b mntiond K-composit stimator has bn usd for yars with som adjustmnts in th CPS - s for instanc Bailar (975, Brau and Ernst (983 or Lnt t al. (994. A complt dscription can b found for instanc in Currnt Population Survy (00. Th adjustmnts known as AK-composit stimator introducd in Gurny and Daly (965 has bn furthr dvlopd,.g. in Cantwll (988 and Cantwll and Caldwll (998. A mor rcnt approach through rgrssion composit stimator has bn considrd in Bll (00, Fullr and Rao (00, Singh t al. (00 (with implications for Canadian Labour Forc Survy. It is basd on modifid rgrssion mthod proposd in Singh (996. Th difficulty in rcursiv stimation in rpatd survys for pattrns with hols was raisd in Yansanh and Fullr (998, who analyzd variancs of composit stimators in svral rotation schms. For a rlativly currnt dscription of th stat of art in th ara on can consult Stl and McLarn (008, in particular Sc. IV on diffrnt rotation pattrns and Sc. V on composit stimators. A vry rcnt papr on optimal stimation undr rotation is by Towhidi and Namazi-Rad (00. In th prsnt papr w dvlop th ida of K-composit stimator in two nw dirctions. First, Q = Q h is allowd to dpnd on th numbr of occasion. Thn it appars that th optimal solution for Q h is vry simpl: it is attaind through minimizing crtain quadratic function F h (which has to b dtrmind on ach occasion. Scond, any rotation pattrn is allowd. Th pric for such a dvlopmnt is surprisingly chap: w only hav to kp track of subsqunt Q h s (to b abl to dtrmin F h s.. Dynamic K-composit stimator W considr a doubl array of random variabls (X i,j which may b column-wis or row-wis infinit or finit, whr th rows ar for valus of th variabl of intrst for diffrnt units on th sam occasion, whil columns ar for valus of th variabl for th sam unit on diffrnt occasions. Thus X i,j rprsnts th valu of th variabl on ith occasion for th jth unit of

11 0 P. Cipila, M. Gniado, J. Wsołowski, M. Wojtyś: Dynamic K-composit... th population. W assum that on a givn occasion all th variabls hav th sam man, which is th paramtr w want to stimat, i.. E X i,j = μ i, i,j =,,... Also it is assumd that thr is no corrlation btwn diffrnt units, i.. Cov(X i,j,x l,k =0 for j = k, i, j, k, l =,,... Ths two assumptions ar crucial for furthr dvlopmnt of our rsult. Th rmaining two ar not important for th drivation w propos but, first, mak formulas somwhat simplr, scond, sinc thy includ som paramtrs which ar assumd to b known, it is dsirabl to hav as fw such paramtrs as possibl. Thus, additionally w assum th xponntial corrlation pattrn btwn th valus of th variabl for th sam unit on diffrnt occasions (which, whil not so important hr, is a crucial condition for th Pattrson schm, i.. Corr(X i,j,x ik,j =ρ k, for any k =0,,..., i,j =,,..., for som ρ [, ]. Finally it is assumd that th variancs of all variabls ar constant, i.. Var X i,j = σ > 0, i,j =,,... Th dynamic vrsion of K-composit stimator, which calld hr DKcomposit stimator, has th form ˆμ h = Q h ( ˆμ h X (h h,h (h X h,h ( Q h X h, h =, 3,... ( whil ˆμ = X, whr all th symbols whr introducd in ( xcpt of Q h which plays th rol of th formr Q. Lt us point out th w do not impos any a priori rstrictions on th rang of (Q h (rstriction imposd in RG on th rang of Q, Q (0,, mad it possibl to pass to th limit with h in th xprssion for th varianc of ˆμ h. Our goal is to choos Q h in a dynamic way, i.. on ach occasion h, th valu Q h has to minimiz th varianc of ˆμ h. Th rotation schm is dscribd by th rotation matrix R =(r i,j, whr r i,j =if th jth unit is in th sampl on th ith occasion, othrwis r i,j =0. Thr is absolutly no rstriction on th rotation pattrn. By n i w dnot th sampl siz on th ith occasion, and m i dnots th siz of ovrlap btwn sampls on occasions (i th and ith. Dnot also for k =, 3,...

12 STATISTICS IN TRANSITION nw sris, March 0 Q i Q i... Q k, for i =,...,k, D i,k =, i = k. Thn w dfin wights which will b rsponsibl for th form of quadratic functions (F k to b minimizd: w ( i,j = n (3 and for any i =,...,k > and any j =,,... [ ( w (k i,j = r ri,j i,j D i,k ( D i,k r ] i,j m i n i n i m i, (4 whr in th last xprssion w adopt th rul that r k,j = r 0,j =0. Not that with such a littl abus of notation th formula for w ( i,j agrs with ( (4. Lt us mphasiz that to find th wights for a givn occasion k w (k i,j nothing mor is ndd but k numbrs Q,...,Q k (not that Q =0,by th dfinition of ˆμ. Now w ar rady to prsnt our main rsult which xplains how to choos, occasion by occasion, th valus (Q h which mak th stimator ˆμ h optimal in th modl w considr hr. Though th formulas, in particular (6, do not look vry frindly, it has to b mphasizd that actually to find Q h on nds just Q,...,Q h to calculat w (h i,j and consquntly, A h, B h and C h. Thorm. In th modl dscribd abov th optimal valu of Q h which minimizs th varianc of DK-composit stimator ˆμ h, h, is C h B h Q h = (5 A h B h C h with C = B and A h B h C h > 0, whr for h A h = σ j h ( i= w (h i,j ( ρσ m h B h = σ n h ri,j j h i= ρ j i <i h w (h i,j h i= w (h i,j w (h i,j r i,jr i,jρ i i r h,j r h,j r i,j ρ h i, (6 w (h i,j r i,j r h,j ρ h i, (7

13 P. Cipila, M. Gniado, J. Wsołowski, M. Wojtyś: Dynamic K-composit... Morovr, ˆμ h = C h = σ n h. (8 h i= j w (h i,j r i,jx i,j with th wights (w (h i,j dfind in (4. Actually, as it will b obsrvd during th proof, which is givn in Sction 3, ( A h B h C h = Var ˆμ h which is always positiv sinc σ > 0. (h (h X h,h X h,h X h 3. Proof Th proof is by induction with rspct to h. For h =th rsult holds tru sinc thn C = B yilds Q =0. Morovr, (4 for k =agrs with th formula for w ( i,j. W assum that it holds for h and w will prov it for h. Comput th varianc of ˆμ h : Var ˆμ h = Q h [ Var ˆμ h Var ( Cov ˆμ h, ( (h (h X h,h Var X h,h Cov ˆμ h, ( ] (h X h,h Cov X(h (h h,h, X h,h (h X h,h [ Q h ( Q h Cov (ˆμ h, X ( h Cov X(h h,h, X ( h Cov X(h h,h, X ] h ( Q h Var X h = Q h A h Q h ( Q h B h ( Q h C h, whr th last quality dfins th quantitis A h, B h and C h. By th induction assumption h ˆμ h = i= j w (h i,j r i,j X i,j. Thn a dirct computation givs

14 STATISTICS IN TRANSITION nw sris, March 0 3 Var ˆμ h = σ j h ( i= w (h i,j ri,j i <i h w (h i,j w (h i,j r i,jr i,jρ i i, Var (h (h X h,h = Var X h,h = σ, m h ( Cov ˆμ h, ( Cov ˆμ h, (h X h,h = σ m h j (h X h,h = σ m h j h i= h i= w (h i,j r i,j r h,j r h,j ρ h i, w (h i,j r i,j r h,j r h,j ρ h i, ( Cov X(h (h h,h, X h,h = σ ρ. m h Combining th last fiv formulas w obsrv that th dfinition of A h agrs with th xprssion (6. Similarly to chck if (7 holds w hav to comput Cov (ˆμ h, X h = σ n h j h i= ( Cov X(h h,h, X h = σ, n h ( Cov X(h h,h, X h = σ ρ. n h Finally, (8 follows sinc Minimizing Var X h = σ n h. w (h i,j r i,j r h,j ρ h i, w gt th solution (5. F h (x =(A h B h C h x (B h C h x C h Th DK-composit stimator is a linar stimator so in gnral it has a form

15 4 P. Cipila, M. Gniado, J. Wsołowski, M. Wojtyś: Dynamic K-composit... ˆμ h = h v i,j r i,j X i,j i= j with som wights (v i,j. To finish th proof w hav to show that v i,j r i,j = w (h i,j r i,j as dfind in (4 for any i =,...,h and any j =,,... Not that by th dfinition of ˆμ h givn in ( w hav ˆμ h = Q h j h i= w (h i,j r i,j X i,j m h j r h,j r h,j X h,j r h,j r h,j X h,j ( Q h r h,j X h,j. m h n h j Comparing th cofficints of X i,j in th last two xprssions w gt for i = h j for i = h v h,j r h,j = Q h r h,j r h,j Q h m h n h ] nh n h = r h,j [ D h,h ( rh,j m h r h,j = w (h h,j r h,j, v h,j r h,j = Q h ( w (h h,j r h,j m h r h,j r hj = Q h r h,j [ D h,h ( rh,j m h = r h,j [ D h,h ( rh,j m h n h and for any i<h [ ( = Q h w (h ri,j i,j r i,j D i,h m i sinc Q h D k,h = D k,h for k = i, i. Thus th proof is compltd. n h ] Q h r h,j r h,j n h m h ] D h,h ( n h r h,j m h v i,j r i,j ( D i,h r ] i,j n i n i m i = w (h h,j r h,j, = w (h i,j r i,j

16 STATISTICS IN TRANSITION nw sris, March Numrical xampls Blow, similarly to numrical comparisons in RG, w considr prcntag gain in fficincy for th DK-composit stimator compard to th man of th obsrvations from th last hth occasion. It is dfind as g h = Var X h Var ˆμ h Var ˆμ h 00. (9 W took h =0, sinc th paramtr Q h bhavs quit stabl with rspct to occasion numbr h. In Tabls and w giv th optimal wight Q h, th varianc of μ h and g h for diffrnt valus of corrlation ρ in two schms: Szarkowski s 00 (Tabl and CPS (Tabl. W can asily s that th largst gain is achivd for strong corrlations and th smallst whn thr is no corrlation btwn occasions for th sam unit. Tabl. Szarkowski schm Tabl. CPS schm ρ Q 0 Var ˆμ 0 g 0 ρ Q 0 Var ˆμ 0 g Sourc: own calculations Considr now a cascad rotation schm which is dfind through a rotation pattrn (,ε,...,ε k,, ε l {0, }, l =,...,k, which movs on unit down th rotation matrix with subsqunt occasions, that is (r i,i,...,r i,ik =(,ε,...,ε k,

17 6 P. Cipila, M. Gniado, J. Wsołowski, M. Wojtyś: Dynamic K-composit... for any i =,,..., othrwis r i,j =0. Th numbr k is calld th rotation pattrn lngth. Taking th advantag of th fact that th DK-composit stimator allows for any rotation schm, w calculatd prcntag gain (as dfind in (9 in fficincy for all possibl cascad schms with rotation pattrns of lngth up to 0. Tabl 3 contains 0 schms with th smallst and 0 with th largst gain among such 8 = 56 schms. Hr, again, th rsults for h =0 ar prsntd. Th largst gain is achivd for "spars" schms with small numbr of lmnts in th rotation pattrn (and strong corrlations whil th lowst gain is obsrvd for schms with complt or almost complt rotation pattrns (and for wak corrlations. Similar comparisons of variancs for particular rotation cascad pattrns in th tim sris framwork can b found in McLarn and Stl (000 (s also Stl and McLarn (00. Tabl 3. Th worst and th bst rotation pattrns worst pattrns ρ g 0 bst pattrns ρ g Sourc: own calculations

18 STATISTICS IN TRANSITION nw sris, March 0 7 Tabl 4. Comparison btwn K-composit and DK-composit stimators ρ Q 0 g 0 diff Q 0 g 0 diff Q 0 g 0 diff m r = m r = m r = m r = m r = Sourc: own calculations

19 8 P. Cipila, M. Gniado, J. Wsołowski, M. Wojtyś: Dynamic K-composit... Tabl 4. Comparison btwn K-composit and DK-composit stimators, continuation Sourc: own calculations ρ Q 0 g 0 diff Q 0 g 0 diff m r = m r = m r = m r = m r = Tabl 4 shows Q h, g h and th diffrnc (diff = g g h btwn th gains obtaind in two ways: g for th K-composit stimator as computd in Tabl ofrgandg h for th DK-composit stimator as proposd in th prsnt papr. W took h = 00 though in many particular cass th valus for Q h and g h stabilizd much arlir. Th numbrs r and m ar rsponsibl for th rotation pattrn, i.. a unit stays in th sampl for r occasions, lavs th sampl for m occasions, coms back into th sampl for r occasions, and so on. Tabl 4 is namd "comparison" nvrthlss w cannot actually in strictly mathmatical sns compar ths two valus of g and g h bcaus th two mthods involv diffrnt modls: RG considrd a finit population cas in which a givn unit rturns to th survy infinitly oftn whras in th prsnt papr an infinit population modl is invstigatd and a unit rturns to th survy aftr a gap of m occasions for anothr squnc of r occasions and

20 STATISTICS IN TRANSITION nw sris, March 0 9 thn lavs th survy. In th cours of simulations w notd that Q h for th DK-composit stimator ar quit stabl, vn for rlativly small valus of h. Morovr, thir valus obsrvd in simulations wr quit similar to thos of Q, obtaind in Tabl of RG. In our Tabl 4 it is visibl that diffrncs in gains of fficincy ar rmarkabl for strong corrlations and small gaps m, whil for small corrlations and larg gaps m thy ar insignificant. Small ngativ valus which appar in som cass ar du to th fact that th two mthods ar not prcisly quivalnt, othrwis ngativ valus would not b possibl sinc th mthod w prsnt hr is optimal within considrd class of stimators. REFERENCES BAILAR, B. (975. Th ffcts of rotation group bias on stimats from panl survys. J. Amr. Statist. Assoc. 70, BELL, P. (00. Comparison of altrnativ Labour Forc Survy stimators. Survy Mth. 7(, BREAU, P., ERNST, L. (983. Altrnativ stimators to th currnt composit stimator. Proc. Sc. Survy Rs. Mth., Amr. Statist. Assoc., CANTWELL, P.J. (988. Varianc formula for th gnralizd composit stimator undr balancd on-lvl rotation plan. SRD Rsarch Rport Cnsus/SRD/88/6, Burau of th Cnsus, Statistical Rsarch Division, -6. CANTWELL, P.J., CALDWELL, C.V. (998. Examining th rvisions in monthly rtail and wholsal trad survys undr a rotation panl dsign. J. Offic. Statist. 4, Currnt Population Survy (00. Dsign and Mthodology, Tchnical Papr 63RV, Burau of Labour Statistics, U.S. Cnsus Burau. FULLER, W., RAO, J.N.K. (00. A rgrssion composit stimator with application to th Canadian Labour Forc Survy. Survy Mth. 7(, GURNEY, M., DALY, J.F. (965. A multivariat approach to stimation in priodic sampl survys. Proc. Amr. Statist. Assoc., Sct. Soc. Statist., HANSEN, M.H., HURWITZ, W.N., NISSELSON, H., STEINBERG, J. (955. Th rdsign of th cnsus currnt population survy. J. Amr. Math. Assoc. 50,

21 0 P. Cipila, M. Gniado, J. Wsołowski, M. Wojtyś: Dynamic K-composit... KOWALSKI, J. (009. Optimal stimation in rotation pattrns. J. Statist. Plan. Infr. 39(4, LENT, J., MILLER, S., CANTWELL, P. (994. Composit wights for th Currnt Population Survy. Proc. Sc. Survy Rs. Mth., Amr. Statist. Assoc., MCLAREN, C.H., STEEL, D.G. (000. Th impact of diffrnt rotation pattrns on th sampling varianc of sasonally adjustd and trnd stimats. Survy Mth. 6(, PATTERSON, H.D. (950. Sampling on succssiv occasions with partial rplacmnt of units. J. Royal Statist. Soc., Sr. B, POPIŃSKI, W. (006. Dvlopmnt of th Polish Labour Forc Survy. Statist. Transit. 7(5, RAO, J.N.K., GRAHAM, J.E. (964. Rotation dsigns for sampling on rpatd occasions. Ann. Math. Statist. 35, SINGH, A.C. (996. Combining information in survy sampling by modifid rgrssion. Proc. Sct. Survy Rs. Mth., Amr. Statist. Assoc., 0-9. SINGH, A.C., KENNEDY, B., WU, S. (00. Rgrssion composit stimation for th Canadian Labour Forc Survy with a rotating panl dsign. Survy Mth. 7, STEEL, D., MCLAREN, C. (00, In sarch of a good rotation pattrn. In: Advancs in Statistics, Combinatorics and Rlatd Aras. Singapor, World Scintific, STEEL, D., MCLAREN, C. (008. Dsign and analysis of rpatd survys. Cntr for Statist. Survy Mth., Univ. Wollonong, Working Papr -08, -3, SZARKOWSKI, A., WITKOWSKI, J. (994, Th Polish labour forc survy. Statist. Transit. (4, TOWHIDI, M., NAMAZI-RAD, M.-R. (00. An optimal mthod of stimation in rotation sampling. Adv. Appl. Statist. 5(, WESOŁOWSKI, J. (00. Rcursiv optimal stimation in Szarkowski rotation schm. Statist. Transit. (, YANSANEH, I.S., FULLER, W. (998. Optimal rcursiv stimation for rpatd survys. Survy Mth. 4, 3-40.

22 STATISTICS IN TRANSITION-nw sris, March 0 STATISTICS IN TRANSITION-nw sris, March 0 Vol. 3, No., pp. 36 ESTIMATION OF POPULATION MEAN USING TWO AUXILIARY SOURCES IN SAMPLE SURVEYS Diwakar Shukla, Sharad Pathak and Narndra Singh Thakur ABSTRACT This papr proposs familis for stimation of population man of th main variabl undr study using th information on two diffrnt auxiliary variabls undr simpl random sampling without rplacmnt (SRSWOR schm. Thr diffrnt classs of stimators ar constructd, xamind with a complt study with othr xisting stimators. Th xprssion for bias and man squard rror of th proposd familis ar obtaind up to first ordr of approximation. Usual ratio stimator, product stimator, dual to ratio stimator, ratio-cum-product typ stimator and many mor stimators ar idntifid as particular mmbrs of th suggstd family. Exprssions of optimization ar drivd and thortical rsults ar supportd by numrical xampls. Ky words: Family of stimators, SRSWOR, Bias and Man squard rror. AMS Subjct Classification: 94A0, 6D05. Introduction To improv th xactitud in sampl survys thory th us of two auxiliary variabls for stimation of population man of a variabl undr study has playd an influntial rol. A numbr of stimators ar accssibl in th litratur of sampl survys whr supporting information is th contributor to improv th mthodology. Out of all ratio and product stimators ar good xampls as vidnc to stat this. Th ratio stimation mthod is practical whn th corrlation cofficint btwn th study and auxiliary variabl is positiv [Cochran (940, 4]. If th corrlation cofficint btwn th study and auxiliary variabl is ngativ thn th us of product stimation will mak th study valuabl [Robson (957 and Murthy (964]. Dpartmnt of Mathmatics and Statistics. Dr. Hari Singh Gour Cntral Univrsity, Sagar, M.P., India sharadpathakstats@yahoo.com, diwakarshukla@rdiffmail.com. Cntr for Mathmatical Scincs, Banasthali Univrsity, Rajasthan, India.

23 D. Shukla, Sh. Pathak, N. Singh Thakur: Estimation of Thr ar so many situations in survy sampling whr th rcord of mor auxiliary variabl is availabl for th invstigators (at last for two variabls. Thr ar so many rsarchrs who usd th information of mor than two auxiliary variabls to contribut in th fild. Dalabhara and Sahoo (994 prsntd a class of stimators in stratifid sampling with two auxiliary variabls for stimation of man. In anothr contribution Dalabhara and Sahoo (000 proposd an unbiasd stimator in two-phas sampling using two auxiliary variabls. Abu-Dayyh t al. (003 usd auxiliary variabls to show stimators of finit population man. Sahoo and Sahoo (993 suggstd a class of stimators in two-phas sampling using two auxiliary variabls. In anothr work Sahoo and Sahoo (00 discussd about prdictiv stimation of finit population man in two-phas sampling using two auxiliary variabls. Singh and Shukla (987 hav a discussion on on paramtr family of factor typ ratio stimator. In a study Shukla t al. (99 transformd factor typ stimator to mak th stimation mor ffctiv. Shukla (00 Studid F-T stimator and sampling procdur undrtakn was two-phas sampling. In this squnc Singh and Singh (99 providd Chain typ stimator with two auxiliary variabls undr doubl sampling schm. In anothr study Singh t al. (994 suggstd a class of chainratio stimator with two auxiliary variabls and th study compltd undr doubl sampling schm. Kadilar and Cingi (004 took two auxiliary variabls in simpl random sampling to find population man. Morovr, Kadilar and Cingi (005 drivd a nw stimator using two auxiliary variabls. Prri (007 analysd th work of Singh (965, 967b and suggstd a nw improvd work on ratio-cumproduct typ stimators with th application of Srivnkataramana, T. (980 stimator on prvious proposd work of Singh (967b. Many authors including Srivastava (97, Srivastava and Jhajj (983, Ray and Sahai (980, Khar and Srivastava (98, Hansn t al.(953 and Dsraj (965 usd mor than on supporting information to mak th study mor imprssiv. Som othr usful contributions ovr applications of auxiliary information ar du to Mukhopadhyay (000, Cochran (005, Murthy (976, Sukhatm t al. (984, Naik and Gupta (99, Singh and Shukla (993 and Shukla t al (009 tc.. Notations and Assumptions Notations for th study ar: Y, X, and X : Population Paramtrs y, x and x : Man pr unit stimats for a simpl random sampl of siz n. n : Sampl siz f : Sampling friction (f = n/n N : Population siz

24 STATISTICS IN TRANSITION-nw sris, March 0 3 ρ 0 : Corrlation btwn variabl Y and X ρ : Corrlation btwn variabl Y and X 0 ρ : Corrlation btwn variabl X and X C SY Y = Y : Cofficint of variation for variabl Y C 0 S X C X = : Cofficint of variation for variabl X C X X X ( S C X = : Cofficint of variation for variabl X C ( 3. Som Estimators In th litratur of survy sampling so many stimators and stimation procdurs xist. This litratur is th basic motivation to work in this dirction and contribution in this ara. Lt Y is th main variabl and X, X ar two auxiliary variabl thn som wll known stimators ar as follows. 3.. Ratio stimator X y R = y (3.a x R R Y = YM[ C X ρ CY C X ] [ C C ρ C C ] Bias( y = E( y (3.b MSE( y R = Y M Y X Y X ; M = (3.c n N 3.. Product stimator x y P = y (3.a X Bias( y = E( y Y = YM ρ C C (3.b P MSE( y R P [ C C ρ C C ] Y X = Y M Y X Y X ; M = (3.c n N 3.3. Dual to ratio stimator [By Srivnkataramana, T. (980] NX nx y VR = y (3.3a ( N n X

25 4 D. Shukla, Sh. Pathak, N. Singh Thakur: Estimation of Y Bias( yvr = E( yvr Y = ρ CY C X (3.3b N MSE( yvr = Y M[ CY α CX αρ CYC X ] ; M =, α = n /( N n n N (3.3c 3.4. Ratio-cum-product typ stimator Singh (965, 967b proposd som ratio-cum-product typ stimators as X x y R = y (3.4a x X R E( yr Y = YM[ C ρ0 C0C ρ0 C0C ρ C ] [ C C C ρ C C ρ C C C ] Bias( y = C (3.4b = Y M MSE( y R ρ C (3.4c X X y R = y (3.5a x x R = E( yr Y [ C C ρ C C ρ C C ρ C ] Bias( y = YM C 0 [ C C C ρ C C ρ C C C ] = Y M MSE( y R ρ C (3.5b (3.5c x x y P = y X X (3.6a P E( yp Y [ ρ C C ρ C C ρ C ] Bias( y = = YM C (3.6b [ C C C ρ C C ρ C C C ] = Y M MSE( y P ρ C (3.6c x X y P = y (3.7a X x P = E( yp Y 0 [ C ρ C C ρ C C ρ C ] Bias( y = YM C 0 0 [ C C C ρ C C ρ C C C ] 0 0 (3.7b MSE( y P = Y M ρ C (3.7c whr M = n N

26 STATISTICS IN TRANSITION-nw sris, March Proposd Estimator(s Singh and Shukla (987 discussd a family of factor-typ (F-T ratio stimator for stimating population man. In anothr contribution Singh and Shukla (993 drivd fficint factor-typ stimator for stimating th sam population paramtr. Driving motivation from both som proposd stimators ar givn blow. ( yf T = ytt T ( yf T = y (4. T T ( yf T 3 = y T ( Ai Ci X i fb xi Whr Ti = (4. ( Ai fbi X i Ci xi A i = ( Ki ( Ki ; Bi = ( Ki ( Ki 4; Ci = ( Ki ( Ki 3( Ki 4 (4.3 Rmark 4. Hr w hav a combination of K i whr i = (,. Som of th factors ar shown in th following tabl whr ( K = K. As abov concrnd K i whr i = (, is constant to choos suitably so that th rsulting man squard rror of proposd stimators may bcom last. For xampl lt K i = X X thn th valus of T and T will b and rspctivly and so on. x x Rmark 4. By proposd stimator w can obtain so many diffrnt stimators. For ach combination of K, an stimator xists. ( K Tabl 4.. Som Mmbrs of th proposd stimation. X t = y x X x = K = (At K x t 5 = y X X x =, K = (At K X t = y x x X =, K = (At K x t 6 = y X x X = K = (At K t t 3 X = y x NX nx ( N n X = (At K =, K 3 7 x = y X NX ( N n X nx = (At K =, K 3 X 4 y x =, K = t = (At K 4 x 8 y X =, K = t = (At K 4

27 6 D. Shukla, Sh. Pathak, N. Singh Thakur: Estimation of Tabl 4.. Som Mmbrs of th proposd stimation (cont.. t 9 NX nx X = y ( N n X x t 0 NX nx x = y ( N n X X t NX nx NX nx = y ( N n X ( N n X t NX nx = y ( N n X (At K =, K 3 = X 3 y x = 4, K = t = (At K At K =, K 3 = x 4 y X = 4, K = t = (At K (At K = K 3 t 5 = NX nx = y ( N n X = 4, K = (At K 3 (At K =, K 4 3 = y (At K = K 4 = 5. Proprtis of Proposd Estimator For larg sampl approximation w assum that fbi y = Y ( 0 ; x = X( ; x = X ( ; α i = ; Ai fbi Ci Ci β i = A fb C i i E = E( = E( 0 ; ( 0 = ( M C δ i = αi E = ; βi i E ( 0 = M C 0 ; ( M C 0 = Mρ0C0 ; E ( 0 Mρ0C0 C E ( C M = n N E = ; = ; E ( = MρC C ; THEOREM 5.: []: Th stimator ( yf T in trms of 0, and up to first ordr of approximation could b xprssd as: [ 0 δ ( 0 β δ ( 0 β δδ ] ( yf T = Y (5. []: Bias of ( yf T up to first ordr approximation is: [ δ ( ρ0c0c βc δ ( ρ0c0c β C δδ ρc ] B( yf T = YM C (5. [3]: Man squard rror of yf T up to first ordr approximation is: [ C0 δ C δ C δρ 0C0C δ ρ0c0c δδ ρc ] = Y M M ( yf T C (5.3

28 STATISTICS IN TRANSITION-nw sris, March 0 7 Proof 5.: []: ( ( ( ( ( x C X fb A x fb X C A x C X fb A x fb X C A y y T F = 0 ( ( ( ( ( ( = Y y T F β β α α [ ] ( ( ( Y y T F δ δ β δ β δ = []: [ ] [ ] } ( ( { ( Y E Y y E T F δ δ β δ β δ = [ ] ( ( ( C C C C C C C C YM y B T F ρ δ δ β ρ δ β ρ δ = [3]: [ ] ] ( ( [ ( Y Y y T F δ δ β δ β δ = ] [ ( C C C C C C C C C M Y y M T F ρ δ δ ρ δ ρ δ δ δ = THEOREM 5.: [4]: Th stimator ( T F y in trms of 0, and up to first ordr of approximation could b xprssd as: [ ] ( ( ( Y y T F δ δ α δ β δ = (5.4 [5]: Bias of ( T F y up to first ordr approximation is: [ ] ( ( ( C C C C C C C C M Y y B T F ρ δ δ ρ α δ β ρ δ = (5.5 [6]: Man squard rror of ( T F y up to first ordr approximation is: [ ] ( C C C C C C C C C M Y y M T F ρ δ δ ρ δ ρ δ δ δ = (5.6 Proof 5.: [4]: ( ( ( ( ( x fb X C A x C X fb A x C X fb A x fb X C A y y T F = 0 ( ( ( ( ( ( = Y y T F α β β α [ ] ( ( ( Y y T F δ δ α δ β δ =

29 8 D. Shukla, Sh. Pathak, N. Singh Thakur: Estimation of [5]: [ ] [ ] ( ( ( E Y Y y E T F δ δ α δ β δ = [ ] ( ( ( C C C C C C C C YM y B T F ρ δ δ ρ α δ β ρ δ = [6]: [ ] ( ( Y y E y M T F T F = [ ] ( C C C C C C C C C M Y y M T F ρ δ δ ρ δ ρ δ δ δ = THEOREM 5.3: [7]: Th stimator 3 ( T F y in trms of 0, and up to first ordr of approximation could b xprssd as: [ ] ( ( ( Y y T F δ δ β δ α δ = (5.7 [8]: Bias of 3 ( T F y up to first ordr approximation is: [ ] ( ( ( C C C C C C C C YM y B T F ρ δ δ β ρ δ ρ α δ = (5.8 [9]: Man squard rror of 3 ( T F y up to first ordr approximation is: [ ] ( C C C C C C C C C M Y y M T F ρ δ δ δ ρ δ ρ δ δ = (5.9 Proof 5.3: [7]: 3 ( ( ( ( ( x fb X C A x C X fb A x C X fb A x fb X C A y y T F = 0 3 ( ( ( ( ( ( = Y y T F α β β α [ ] ( ( ( Y y T F δ δ β δ α δ = [8]: [ ] ( ( ( Y y T F δ δ β δ α δ = [ ] ( ( ( C C C C C C C C YM y B T F ρ δ δ β ρ δ ρ α δ =

30 STATISTICS IN TRANSITION-nw sris, March 0 9 [9]: E [( y Y ] = E[ Y { δ ( α δ ( β δ δ }] F T [ C δ C δ C ρ C C δ ρ C C δ δ δ ρ C ] F T 3 = Y M M ( y C 6. Minimum Man Squard Error & Optimal Choics for Proposd Estimator(s In this proposd stimator w hav multipl choics of th combination K i ; i = (, and optimal conditions obtaind by man squard rror of all proposd dsigns. For minimum man squard rror by ( yf T diffrntiating (5.3 with rspct to δ and δ rspctivly and quating to zro. C δ C C ρ δ ρ C C ρ C C δ C 0 δ ρ C C = 0 = 0 By solving ths simultanous quations, w hav (6. C0 ρ0ρ ρ0 δ ˆ = = δ C ( ρ C0 ρ0ρ ρ0 and δ ˆ = = δ C ( ρ (6. At ths valus of ˆ δ ˆ and δ th minimum man squar rror of th proposd stimator is [ V ( V ρ U ( U ρ UV ] F T = Y C0 M 0 0 Min MSE( y ρ (6.3 ρ0ρ ρ whr U = ( ρ 0 and V ρ = 0 ρ ( ρ ρ 0 By adopting th sam procdur w can obtain th minimum man squard rror corrsponding to ( yf T and ( yf T 3 by (5.6 and (5.9.

31 30 D. Shukla, Sh. Pathak, N. Singh Thakur: Estimation of Th information of optimization rgarding ( yf T and ( yf T 3 is ˆ δ = ˆ δ ; ˆ δ = ˆ δ ; ˆ δ = ˆ δ and ˆ δ = ˆ δ (6.4 Rwriting (6., as ˆ C δ = C ˆ δ 0 C = C 0 ρ ρ 0 ρ ( ρ 0 ρ ρ ρ ( ρ 0 0 = = ( say ( say (6.5 From (6.5 w can obtain th rlation in th form of charactrizing scalar as follows ( ( K 3 K 3 ( f ( f f f K 9 K (3 5 f 5 f 6 K (4 f 4 f 4 = 0 ( f f K (4 f 4 f 4 = 0 (6.6 Abov polynomial (6.6 provids thr choics of Kand K for th minimum man squard rrors of proposd stimators. In th similar way ˆ δ = ; ˆ δ ˆ ˆ = ; δ = andδ = will also provid th polynomials of dgr thr i.. in ach cas w hav thr diffrnt choics of constant K i ; i =, to improv th stimator. 7. Empirical study Th targt in this sction is to valuat th gain in fficincis (in trms of ms obtaind by th proposd stimators. To s th prformanc of th various stimators discussd hr, w ar considring two diffrnt population data usd arlir by othr rsarchrs. Th mpirical analysis is discussd blow. Population - [sourcs: Andrson (958] y : Had lngth of scond son x : Had lngth of first son x : Had bradth of first son

32 STATISTICS IN TRANSITION-nw sris, March 0 3 Th rquird information is givn in Tabl 7.. Tabl 7.. Population Paramtrs. Paramtr Valu Paramtr Valu Paramtr Valu Paramtr Valu Y n 7 C ρ X 85.7 N 5 C ρ ρ X 5. f 0.8 C Tabl 7.. Prcnt Rlativ Efficincy of various stimators with rspct to man pr unit stimator for Population. Estimator(s PRE ( with rspct to y ( yf T ( yf T ( yf T 3 y t t t t t t t t t t t t t t t * ( yf T * ( yf T * ( yf T

33 3 D. Shukla, Sh. Pathak, N. Singh Thakur: Estimation of Population - [sourcs: Stl and Torri (960, p.8] y : Log of laf burn in sc x : Potassium prcntag x : Chlorin prcntag Th information rgarding population - is givn in Tabl 7.3. Tabl 7.3. Population - Paramtrs. Paramtr Valu Paramtr Valu Paramtr Valu Paramtr Valu Y n 6 C ρ X N 30 C 0.95 ρ ρ X f 0.0 C Tabl 7.4. Prcnt Rlativ Efficincy of various stimators with rspct to man pr unit stimator for Population. % RE ( with rspct to y Estimator(s ( yf T ( yf T ( yf T 3 y t t t t t t t t t t t t t t t * ( yf T * ( yf T * ( yf T

34 STATISTICS IN TRANSITION-nw sris, March Discussion & Conclusion For population- th choics to optimization of man squard rror of ( yf T can b drivd from (6.5 which giv a polynomial of dgr thr (6.6. On solution w hav [ K ] = ; [ K ] = ; [ K ] 3 =. 65 ; [ K ] = ; [ K ] =. 985 and[ K ] 3 = For ( y F T th valus ar[ K ] = [ ] 4 K, [ K ] [ ] 5 = K ; [ ] [ ] K 6 = K 3 and [ K ] 4 =. 93. Similarly for ( yf T 3 valus ar[ ]. 906 K = whras K 7 = ; [ K ] = [ ] 7 K ; [ K ] = [ ] 8 K and [ ] [ ] 9 K 3 othr roots ar imaginary. For population- th choics of th constant scalar squard rror of ( yf T ar 39.. K i to rduc th man [ K ] 95 ; [ K ] 5859 ; [ K ]. 097 and [ ]. 939 = 4 = = 3 = K. For = 5 K K = ; ( y F T th valus ar[ K ] = [ ] 4 K, [ K ] = [ ] ; [ ] [ ] 6 [ ] 3 K ; [ K ]. 855 and [ ] = valus ar[ ] * 7 = * 6 = 7 K K = K and [ ] [ ] ( yf T, ( yf T and ( yf T to man pr unit stimator in th abov mntiond tabls. * 3 K ( yf T K. Similarly for 3. Th rmaining roots ar imaginary. dnots th optimal fficincy gain with rspct From ths rsults it is crtain that th proposd stimators submit a wid ground for th optimization by multipl choics of th charactrizing scalar K i. Sinc th gnration of th stimators by th proposd classs is asy, a numbr of stimators can b abl to achiv for mor study. Th proposd stimator proposd a wid choic for th charactrizing scalar, which is th bauty of th proposd analysis. By th compilation of th prcntag rlativ fficincis corrsponding to population- and shown in tabl-7. and tabl-7.4 it is clar that th proposd stimators ar mor fficint than th othr xisting stimators as ratio stimator, product stimator, dual to ratio stimator, man pr unit stimator, ratio-cumproduct typ stimator, tc., and many mor chain typ stimators which ar discussd abov, with considrabl gain in trms of man squar rror. Thus, th proposd stimators ar rcommndd for us in practic.