SURVEY SAMPLING IN ECONOMIC AND SOCIAL RESEARCH

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1 SURVEY SAMPLIG I ECOOMIC AD SOCIAL RESEARCH

2 Studa Eonomczne ZESZYTY AUKOWE WYDZIAŁOWE UIWERSYTETU EKOOMICZEGO W KATOWICACH

3 SURVEY SAMPLIG I ECOOMIC AD SOCIAL RESEARCH Edted b Janusz L. Wwał and Tomasz Żądło Katowce 0

4 Komtet Redacn Kstna Lseca (pzewodncząca, Anna Lebda-Wbona (seetaz, Floan Kuźn, Maa Mchałowsa, Anton edelńs, Iena Pa, Stansław Swadźba, Tadeusz Tzasal, Janusz Wwał, Teesa Żabńsa Komtet Redacn Wdzału Zaządzana Janusz Wwał (edato naczeln, Wocech Gamot (seetaz, Teesa Żabńsa, Jace Szołtse, Włodzmez Rudn Rada Pogamowa Loenzo Fatton, Mao Glow, Mloš Kál, Bonsław Mcheda, Zdeně Moláš, Maan oga, Gwo-Hsu Tzeng Redato Elżbeta Spadzńsa-Ża Copght b Wdawnctwo Unwestetu Eonomcznego w Katowcach 0 ISB ISS Wesą pewotną Studów Eonomcznch est wesa papeowa Wszele pawa zastzeżone. Każda epoduca lub adaptaca całośc bądź częśc nnesze publac, nezależne od zastosowane techn epoduc, wmaga psemne zgod Wdawc WYDAWICTWO UIWERSYTETU EKOOMICZEGO W KATOWICACH ul. Maa 50, Katowce, tel , fa e-mal:

5 COTET ITRODUCTIO... 7 Czesław Domańs FIRST ASSOCIATIOS OF POLISH STATISTICIAS... Steszczene... 7 Wocech Gamot O POOL-ADJACET-VIOLATORS ALGORITHM AD ITS PERFORMACE FOR O-IDEPEDET VARIABLES... 8 Steszczene... 9 Anna Imołe, Janusz Gołaszews, Dausz Załus, Zbgnew asals PRACTICAL STATISTICAL AD ECOOMIC ASPECTS OF USIG SURVEY STUDIES FOR IDETIFICATIO OF THE KEY PLAT CULTIVATIO TECHOLOGY FACTORS... 3 Steszczene Aadusz Kozłows THE USEFULESS OF PAST DATA I SAMPLIG DESIG FOR EXIT POLL SURVEYS Steszczene Jan Kubac, Alna Jędzecza THE COMPARISO OF GEERALIZED VARIACE FUCTIO WITH OTHER METHODS OF PRECISIO ESTIMATIO FOR POLISH HOUSEHOLD BUDGET SURVEY Steszczene Doota Raczewcz SOME ASPECTS OF POST EUMERATIO SURVEYS I POPULATIO CESUSES I POLAD AD GERMAY Steszczene... 76

6 Ondře Vlus OPTIMIZATIO OF SAMPLE SIZE AD UMBER OF TASKS PER RESPODET I COJOIT STUDIES USIG SIMULATED DATASETS Steszczene Janusz L. Wwał O LIMIT DISTRIBUTIO OF HORVITZ-THOMPSO STATISTIC UDER THE REJECTIVE SAMPLIG Steszczene Tomasz Żądło O ACCURACY OF TWO PREDICTORS FOR SPATIALLY AD TEMPORALLY CORRELATED LOGITUDIAL DATA Steszczene AUTHORS... 06

7 ITRODUCTIO Sample suves povde one of the most challengng felds fo applng the statstcal methodolog. The confont the eseache wth a vast dvest of unque pactcal poblems encounteed n the couse of studng populatons. The nclude, but ae not lmted to: non-samplng eos, specfc populaton stuctues, contamnated dstbutons of stud vaables, non-satsfacto sample szes, ncopoaton of the aula nfomaton avalable on man levels, smultaneous estmaton of chaactestcs n vaous subpopulatons, ntegaton of data fom man waves o phases of the suve and ncompletel specfed samplng pocedues. Omnpesent constants on tme and cost addtonall complcate the pocess of desgnng a suve. Dealng wth such condtons bngs about the need fo fomulatng sophstcated statstcal pocedues dedcated to specfc condtons of a sample suve. It gves bth to wde vaet of appoaches, methodologes and pocedues boowng the stength fom vtuall all banches of statstcs. Ths monogaph was pepaed on the bass of the papes that wee pesented dung the seventh confeence Suve Samplng n Economc and Socal Reseach that too place on 8-0 Septembe 0 n Katowce, Poland. The chaptes ae etended and mpoved vesons of the confeence papes. The authos deal wth vaous theoetcal and pactcal ssues. The common motve of all papes s the elaton to sample suves. The pape of Czesław Domańs s devoted to the fst thee most valuable achevements of Polsh statstcans and the nfluence on the development of ntenatonal statstcs. Fstl, the Autho dscusses the ole of Tadeusz Plat ( pofesso of statstcs and admnstaton at the Lvov Unvest as a co-founde of the Intenatonal Statstcal Insttute. Secondl, the dea of the fst Polsh census conducted n 789 s pesented, whch was devsed and eecuted b a membe of Palament Fde Józef eal Moszńs ( Thdl, the establshment of the fst Polsh Cha of Statstcs n the Wasaw School of Law and Admnstaton s dscussed, whch was headed b Wawznec Suowec ( pofesso of statstcs and economcs. Wocech Gamot concentates on the Pool-Adacent-Volatos algothm sometmes abbevated as PAVA. The ognal algothm s fomulated unde the assumpton of ndependence between andom vaables whose epectatons ae to be estmated. Seveal modfcatons of ths pocedue wee developed n the lteatue but unde ndependence. Hence, n ths pape a smulaton stud s caed out to assess popetes of PAVA-based odeed pobablt estmates unde coelaton.

8 8 ITRODUCTIO Anna Imołe, Janusz Gołaszews, Dausz Załus and Zbgnew asals dscuss pactcal statstcal and economc aspects of usng suve studes fo dentfcaton of the e plant cultvaton technolog factos. The consde suve stud caed out n 008 n ode to detemne the e elements n a plant poducton technolog and to calculate unt poducton costs of gowng wnte e (Secale ceeale L. fo gan. The suves coveed e gan poduces n notheasten Poland, who gow e on an aceage of ove ha. The economc analss was pefomed based on dect outlas on poducton; unt costs and dect magn wee calculated and the stuctue of costs as well as poftablt of wnte e poducton wee detemned. Aadusz Kozłows studes usefulness of past data n samplng desgn fo et poll suves. The man stess s put on the use of wdel avalable databases contanng detals of past electons esults. B means of smulaton epements the effectveness of technque of connectng the selecton of new sample wth past esults (ted sample pocedue s evaluated and optmal paametes fo ths technque ae ndcated. A modfcaton of the pocedue s also poposed. The best esults ae obtaned fo statfed samplng wth the use of elements of ted sample pocedue. The possbltes of cost educton of suves wthout peudce to the effectveness b means of the ght selecton of solel lage pecncts ae also ndcated. Jan Kubac and Alna Jędzecza compae Genealzed Vaance Functon wth othe methods of pecson estmaton fo Polsh Household Budget Suve. A statng pont was the estmaton of Balanced Repeated Replcaton vaances o bootstap vaances n the stuaton whee usng BRR was not applcable. To evaluate the GVF model the hpebolc functon was used. The computaton was done usng WesVAR and SPSS softwae and some specal pocedues pepaed fo R-poect envonment. The assessment of estmates consstenc fo countes was also conducted b means of small aea models. Doota Raczewcz pesents some aspects of post enumeaton suves n populaton censuses n Poland and Geman. The Autho begns wth compason of populaton censuses n Poland and Geman. et attenton s pad to data qualt and potental eos n populaton censuses. Compason s made of pncples of post-enumeaton suves n censuses n Poland and Geman. What s moe, ntenatonal ecommendaton on qualt assessment of populaton censuses accodng to the U and EUROSTAT s pesented. Ondře Vlus dcussed optmzaton of sample sze and numbe of tass pe espondent n conont studes usng smulated datasets. The Autho pesents an appoach based on analzng batches of smulated datasets wth gven chaactestcs. The atcle ncludes ovevew of the esults fo choce-based conont studes wth usual level of complet. Seach fo an optmal combnaton

9 ITRODUCTIO 9 of sample sze and numbe of tass pe espondent that allows us to acheve equed accuac of ou outputs wth optmal cost s of man focus but senstvt of the ecommendatons wth espect to changes n fed paametes of the datasets s also ncluded. Janusz L. Wwał studes lmt dstbuton of Hovtz-Thompson statstc unde the eectve samplng. On the bass of the papes b Bege and Snne (005 and Háe (964 he consdes the lmt dstbuton of H-T statstc standadzed b ts sample vaance. Moeove, the vaance of the H-T estmato s consdeed unde the assumpton that the aula vaable value s the obsevaton of the vaable unde stud but wth measung eo. Tomasz Żądło compaes accuac of two pedctos fo spatall and tempoall coelated longtudnal data based on Monte Calo smulaton stud usng R pacage. The fst pedcto unde stud s the empcal best lnea unbased pedcto (EBLUP deved fo some specal case of the Geneal Lnea Med Model whee spatal and tempoal coelatons ae taen nto account. The second pedcto s EBLUP deved unde the assumpton of lac of spatal and tempoal coelaton.

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11 Czesław Domańs FIRST ASSOCIATIOS OF POLISH STATISTICIAS. Intal Remas The ea 0 wll wtness the celebatons of the 00th annvesa of foundng the Polsh Statstcal Socet the fst assocaton of statstcans n Poland. The pesent confeence offes an ecellent oppotunt to pesent the contbuton made b Polsh statstcans towads an oveall development of statstcs as a dscplne of scence and ddactcs. The confeence wll also commemoate the ublee of the Cha of Statstcs whch was establshed 60 eas ago, fst as a pat of the Hghe School of Economcs, then the Academ of Economcs and pesentl the Economc Unvest n Katowce. It s woth pesentng hee thee most valuable achevements of Polsh statstcans and the nfluence on the development of ntenatonal statstcs, long befoe the Polsh Statstcal Socet came nto estence.. Conductng n 789 the fst-eve natonal census, whch was devsed and eecuted b a membe of Palament Fde Józef eal Moszńs ( Establshng n 8 the Cha of Statstcs n the Wasaw School of Law and Admnstaton, whch was headed b Wawznec Suowec ( pofesso of statstcs and economcs. 3. Co-foundng the Intenatonal Statstcal Insttute b pofesso of statstcs and admnstaton at the Lvov Unvest Tadeusz Plat ( Statstcal congesses.. Statstcal Congesses Befoe the Intenatonal Statstcal Insttute was founded n 840 t became customa fo scholas epesentng one banch of scence to meet at congesses whose man am was develop methods of posng and solvng mpotant poblems. The fst dea of statstcal congesses was conceved n London n 85 b Lambet Adolphe Jacques Quételet ( , the

12 Czesław Domańs chaman of the cental Belgan Statststcal Commsson. The congesses wee held n co-opeaton wth the govenment of the espectve count whee the net sesson was scheduled. Standadzng of admnstatve statstcs and adoptng a geneal methodolog of conductng censuses wee the man poblems dscussed dung subsequent statstcal congesses. Altogethe nne sessons of such congesses wee convened n the followng ctes: Bussels (853, Pas (855, Venna (857, London (860, Beln (863, Floence (867, the Hague (869, St. Petesbug (87, Budapest (876. At St. Petesbug Congess n 87 a commsson was set up n ode to supevse wos on the unfcaton of the ntenatonal statstcs. In 878 the commsson poposed that a specal ntenatonal bod, fosteng coopeaton between natonal statstcal offces should be fomed. Howeve, the poposal was eected b Geman and Swtzeland who wee of the opnon that such a bod would unnecessal ntefee wth the ntenal affas of ndvdual countes. As a esult of ths opposton congesses ceased to be convened, et the dea of mang statstcs an ntenatonal dscplne was not abandoned. Followng a sees of meetngs n Pas and London, a conventon was held on June 4, 885 n London, whee the Intenatonal Statstcal Insttute was bought nto beng. The conventon, attended b patcpants, was called sesson and fom that tme onwads all the egula meetngs of membes have become nown as sessons of the Intenatonal Statstcal Insttute (ISI. Polsh statstcans wee among those statstcans who too an actve pat n foundng the Intenatonal Statstcal Insttute and wee the Insttute membes. The most emnent epesentatve of the Polsh statstcs of the tme was August Ceszows ( A. Ceszows, who attended the Statstcal Congess n Pas n 855, was the man speae of one of the sessons of the Congess. The name of Tadeusz Zgmunt Plat should also be bought hee, as the fst Polsh membe of the Insttute among ts 00 statuto epesentatves. Tadeusz Plat ( gaduated fom the Unvest of Lvov and and contnued hs educaton n Beln whee he specalzed n statstcs unde the scentfc supevson of a enowned schola d Edmund Engel. In 867 he eaned hs docto s degee n law on the bass of the doctoal dssetaton Pactce n all poltcal and legal slls. Two eas late Plat obtaned a postdoctoal degee havng wtten the habltaton thess enttled Uebe den Begff des wtschaftlchen Wethes and became a pvate assocate pofesso n the Cha of Socal Economcs of the Lvov Unvest. He was confeed hs second postdoctoal degee n admnstatve law (870 and developed futhe hs nowledge of statstcs dung hs studes n the Pussan Statstcal Offce n Beln (87. In 87 Tadeusz Plat was apponted assstant pofesso of Unvest of Lvov and became the head of the Cha of Statstcs

13 FIRST ASSOCIATIOS OF POLISH STATISTICIAS 3 and Admnstaton, and a few eas late (878 he became a full pofesso. He was elected to be the dean of the Facult of Law and Admnstaton (fou tems of offce.e. 880/88, 884/885, 889/890, 900/90, the Pesdent of the Unvest (886/887, the Vce-Pesdent (887/888. Snce 909 he was an honoa pofesso. In the eas Plat seved seveal tems of offce as a deput fo the Galca Palament. He headed the Statstcal Offce of the atonal Depatment n Lvov fo ove fou decades (874-90, and woed as a deput mashal of the atonal Depatment ( In 888 he was apponted a coespondng membe of the Academ of Leanng n Cacow, and snce 98 a membe of the Polsh Academ of Leanng. He was also a membe of the Boad ( and vce-pesdent of the Galcan Economc Socet, a coespondng membe of the Cental Statstcal Commsson n Venna (snce 876, and a full membe of the Scentfc Socet n Lvov (snce 90. The etensve scentfc output of the Autho ncludes nte ala: Methods of Collectng Data fo Havest Statstcs (87, On Muncpal Statstcal Offces (87, Composton of Commune Repesentaton n Ctes and Towns of Galca (874, Statstcal Pesentaton of Communal Stuctue n Galca and Results of Local Electons (874, The Tetboo of Statstcs of Galca (900, and Statstcs (93. In hs studes Plat pesented methodologcal poblems, focusng on agcultual statstcs. Among methods used n statstcs of vegetable poducton he poposed estmaton methods as an mpotant souce of statstcal nfomaton. 3. Fst Geneal Census n Poland The begnnngs of statstcal actvtes on the Polsh tetoes concde wth the poceedngs of the so called Fou Yea Palament Sesson.e. the eas The Palament adopted a esoluton on cang out n 789 the fst natonal populaton census combned wth smoe egstaton. The Census esults wee to help the Palament to pass a bll on mposng a new ta, whch was supposed to povde mone towads epenses on pemanent, one-hunded- -thousand am. The autho of the statstcal tables of the 789 Census and a statstcal method of the mlta ta calculaton was a deput Fde Józef eal Moszńs ( It s woth notng that although the populaton and smoe egstaton of 789 was the fst state census, numeous othe egstes, nventoes and censuses appeaed n Poland as eal as 6th centu (e.g. populaton census of the Cacow Docese ( and the Ploc Docese of the eas 773, 776 and 778 and the wee conducted fo ta, economc, mlta and chuch easons. The numecal data contaned n these egstes

14 4 Czesław Domańs eman valuable souce of statstcal nfomaton fo all nds of estmatons and analses. Moszńs ponted out that the wealth of the state cannot be measued b the affluence of seveal astocatc famles and a couple of thousands of ch ctzens, but t athe should be measued b settlements, the wealth of townspeople and countmen, pospeous tade and floushng cafts. The statstcal method of the mlta ta assessment poposed b Moszns n the Palament was of absolutel unque chaacte and t was used nowhee else ethe befoe o afte that tme. In ode to measue the value of land and popet n a gven povat (dstct n an obectve wa he poposed a statstcal method based on the followng data: value of land and popet n the povat assessed on the bass of deeds of sales fo the peod of last eas as ecoded n dstct boos; t was seen as epesentatve enough fo mang calculatons, numbe of smoes obtaned fom teasue taffs; both alenated fo the peod of last eas and those whch wee not subect to puchase o sale. The obtaned nfomaton allowed the Teasue Commssons to mae calculatons based on the value of alenated goods, thans to smoe numbes, and assess pecsel the value of popetes n a gven povat tang nto account the popoton between smoes of alenated goods and the total. Fde Józef Moszńs suppoted the dea of the so-called co- -equaton that s a fa fscal sstem whch made the gent and the chuch pa taes based on pofts deved fom the land and popet. As a esult of Moszńs s ntensve effots the Fou Yea Palament ( passed a legslatve act of geat mpotance and etemel nteestng fo the hsto of statstcs n Poland a consttuton of June, 789 nown as Smoe Inspecton and Populaton Regste whch was n fact the fst populaton census n the hsto of Poland. The census ncluded the ual and the uban populaton and ecluded the gent and the cleg. It encompassed the followng categoes: se, occupaton and socal status obsevng the dffeence between sons (n two age goups up to 5 eas and above 5 eas and daughtes. The fst estmatons of the populaton numbe n Poland wee poduced b some of the above-mentoned statstcans. Józef Wbc povded n 777 the estmated numbe of populaton of people; Alesande Buchng n 77 gave the numbe of 8.5 mllon; Stansław Staszc n 785 estmated the populaton at the level of 6 mllon; Fde Moszńs, afte the fst census of 789 poduced the numbe of people; the fgue dd not nclude the gent and the cleg, who wee not the subect of the census, et the numbe was estmated at the level of thousand.

15 FIRST ASSOCIATIOS OF POLISH STATISTICIAS 5 4. Fst Cha of Statstcs The gowth of nteest n statstcs n Poland was maed b the foundaton of the fst cente of statstcal nowledge the Cha of Statstcs at the Wasaw School of Law and Admnstaton n 8. Headng the Cha was entusted to a statstcan and economst Wawznec Suowec ( Afte the fst Cha of Statstcs had been establshed, thee was an upsuge n the nteest n statstcs as a sepaate banch of scence and not as a tool to be used n admnstaton. It s woth tang a close loo at the pofle of the fst Polsh pofesso of statstcs. He was bon n Gnezno povnce n a gent faml of modeate means. Havng completed hs studes Suowec stated pofessonal caee as a pvate tuto what helped hm to get to now academc centes of Venna and Desden. In 807 he became a membe of the Wasaw Socet of Fends of Scences and at the tme he alead bult hs eputaton as an epet n socal ssues. Although he onl gave lectues n statstcs fo one ea, he was engaged n educatonal and scentfc actvtes fo most of hs lfe. In 8 he was apponted as a seceta geneal n the Mnst of Educaton and esgned fom pedagogcal dutes. In the Congess Kngdom of Poland he too the post n the Councl fo admnstatve affas and educatonal funds. The lst of most mpotant studes of W. Suowec s qute long and ncludes among othes: On the fall of ndust and towns n the old Poland (80, On ves and floatng of the Gand Duch of Wasaw (80, On statstcs of the Gand Duch of Wasaw (8-83. Suowec also too nteest n the populaton poblems and eamned dffeent easons fo populaton development. As a statstcan he peceved was, llteac, non-poductve calamtes and maladmnstaton as factos whch advesel affected populaton development. Wawznec Suowec togethe wth Ignac Stawas and Domn Ksńs wee the fst Polsh scholas to defne the subect and tass of statstcs. I. Stawas and D. Ksńs epessed the vews at the meetngs of the Wasaw Socet of Fends of Scences. The fome voced hs opnons n Septembe 809 but hs pesentaton was publshed n the Annals of the Socet n 8 whle the latte pesented them n Apl 84. W. Suowec not onl ceated a boad famewo fo statstcs but also woed to develop ths elatvel new feld of scence b gvng lectues n the Academ of Law and Admnstaton n the academc ea 8/8.

16 6 Czesław Domańs 5. Fnal Remas Two sessons of the Intenatonal Statstcal Insttute wee held n Poland. In August 99 Wasaw was the host of the 8th Sesson of the Intenatonal Statstcal Insttute. The fact that Poland was the oganze of the sesson epessed espect of the ntenatonal statstcal communt fo the achevements of Polsh statstcs. S Polsh epesentatves gave pesentatons n the couse of the Sesson: E. Sztum de Sztem: Statstcal method fo eamnaton of ndces of economc development; E. Lpńs: Remas on wong methods of the Polsh Insttute of Economc and Pce Reseach; S. Rzepewcz: On possblt of compang cme statstcs n dffeent countes; S. Szulc: On the so-called standadzaton o mpovng coeffcents; J. eman: Contbuton to the theo of elablt of statstcal hpotheses; J. Peałewcz: Ependtue and evenue of publc-legal assocatons. In Septembe 975 the 40th Sesson of the Intenatonal Statstcal Insttute was oganzed n Wasaw. Dung the Sesson Polsh statstcans pesented the followng papes: W. Maceews, W. Welfe: Foecastng models fo the natonal econom plannng and the elevance of the natonal nfomaton sstem; K. Zagós: Soco-demogaphc statstcs n the sstem of cental socoeconomc plannng; R. Batoszńs: A model fo s of abes; T. Walcza: The ole of moden statstcal nfomaton sstem fo management and plannng; E. Kzeczowsa: An ntegated sstem of ntenatonal statstcal compasons; J. Kudca: The possbltes of applng nput-output elatons to the econometc macomodels. Refeences Domańs, Cz. (0 Setna ocznca powstana Polsego Towazstwa Statstcznego (00 eas of the Polsh Statstcal Socet. Wadomośc Statstczne n 9(604, wzeseń 0, -0. Domańs, Cz. (004 Jubleusz Polsego Towazstwa Statstcznego Tadce obecne zadana statst w Polsce (Jublee of the Polsh Statstcal Socet. Tadton and the pesent-da tass of statstcs n Poland, ed. A. Zelaś, Wdawnctwo AE Kaów.

17 FIRST ASSOCIATIOS OF POLISH STATISTICIAS 7 Kleczńs, J. (886 Mędznaodow Insttut Statstczn (Intenatonal Statstcal Insttute. Pzegląd Pols oczn XI, t. II, Łuaszewcz, J. (995 Polsa hstoogafa a statsta (Polsh hstoogaph and statstcs. Bblotea Wadomośc Statstcznch, t. 46, Romanu, K. (975 Udzał Pols w pacach Mędznaodowego Insttutu Statstcznego (Contbuton of Poland to the wo of the Intenatonal Statstcal Insttute. Wadomośc Statstczne n 8. PIERWSZE ZRZESZEIE POLSKICH STATYSTYKÓW Steszczene W 0 ou pzpada 00. ocznca powstana pewszego zzeszena statstów polsch Polsego Towazstwa Statstcznego. Wato na te onfeenc zaznaczć wład statstów polsch w ozwó statst ao dscpln nauowe ddatczne. Konfeenca ta ma ówneż chaate ubleuszow, zwązan z 60-lecem Kated Statst, dzałaące wcześne w amach Wższe Szoł Eonomczne, potem Aadem Eonomczne, a obecne Unwestetu Eonomcznego w Katowcach. Wspomnm edne o tzech osągnęcach polsch statstów, tóe maą oddzałwane mędznaodowe:. Pzepowadzene w 789 ou pewszego spsu państwowego, tóego pomsłotwócą głównm ealzatoem bł poseł h. Fde Józef Moszńs ( Powołane w 8 ou Kated Statst w Szole Pawa Admnstac w Waszawe, tóe eownctwo powezono pofesoow statst eonom Wawzńcow Suowecemu ( Współudzał w twozenu Mędznaodowego Insttutu Statstcznego pofesoa statst admnstac Unwestetu Lwowsego Tadeusza Plata (

18 Wocech Gamot O POOL-ADJACET-VIOLATORS ALGORITHM AD ITS PERFORMACE FOR O-IDEPEDET VARIABLES. Intoducton Estmaton of odeed epectatons s a poblem that has attacted attenton of eseaches fo moe than fft eas. Ths nteest s eflected b a wde lteatue statng wth papes of Ae at al (955, Bun (955 and van Eeden (956, 957, 958, Katz (963 as well as Hanson et al (973, Sacowtz and Stawdeman (974 and then developed among othes b Sacowtz (98, Lee (983, Best and Chaavat (990, Chaas and van Eeden (99, Qan (99, Bloc et al (994, Ahua and Oln (00, Budaov et al (004, Jewel and Kalbflesch (004 and Hansohm (007. A good summa of the state of nowledge s pesented n monogaphs of Robetson et al. (988 and van Eeden (006. Somehow, t appeas that appoaches of all these authos shae a common featue. amel, t s alwas assumed that andom vaables fo whch one deses to compute estmates satsfng odeng constants ae ndependent. Ths autho s not awae of an estmaton pocedue that accounts fo possble coelaton between such vaables. Hence two possble choces ae: constuctng a pocedue dedcated to coelated data o nvestgatng the popetes of estng estmaton stateges when ndependence assumpton s dopped. In ths pape the ognal Pool-Adacent-Volatos algothm (PAVA of Ae et al (955 s evsted and ts popetes n the case of non-zeo coelaton ae assessed n a smulaton stud.. Pool-Adacent-Volatos algothm Let,,..., n be unnown pobabltes satsfng a smple ode:... n (

19 O POOL-ADJACET-VIOLATORS ALGORITHM 9 Let ndependent tals be made of an event wth pobablt fo,...,n. Let denote the numbe of successes n the -th tal and let * p / fo,...,n. The PAVA pocedue computes estmates p,..., p n of,..., n satsfng ( b teatvel goupng (megng ntal estmates * * p,..., p n nto blocs and aveagng them wthn each bloc. The pocedue wos though epeatng followng steps (see Hädle (99, Ae et al (955 and de Leeuw et al (009. Assgn each component fo,...,n to a sepaate goup so ntall (0 (0 n goups G,..., G n est. Set ntal estmate of mean pobablt n each -th (0 * goup to ~ p g p fo g,...,n g Whle thee est some goups n the -th step of algothm such that assocated estmates of mean pobablt volate the odeng constant, fnd ( ( ( mamum-length sequence of such goups (sa G g, Gg+,..., Gh and mege them (+ ( ( ( (+ ( nto a sngle goup G g G g G g+... G h whle G G + hg + fo >g and assgn a mean pobablt estmate ~ ( p to the goup ( G + g ( ( whle ~ + p ~ p fo >g + hg g ( + ( + G g G g 3 When teaton stops afte the last sa K-th step (whee K {0,,...}, wth H goups emanng assgn a mean pobablt estmate computed fo a goup to each of ts membe components so that the fnal estmate fo the component s ( p ~ p g fo G g, g,,...,h If,..., n ae ndependent, ths pocedue leads to a vecto of estcted mamum lelhood estmates fo pobabltes,,..., n. We wll now abandon the ndependence assumpton and allow fo some coelaton among s. 3. A smple coelaton model To nvestgate popetes of PAVA estmato n the case when vaables ae coelated we wll assume a smple model statng that coelaton coeffcent between ndvdual bna vaables s the same fo all pas of subsequent vaables: (,, (, 3, ( 3, 4,...,( -,. Hence a pocedue geneatng bna andom vectos n the fom [,..., ] satsfng E( m and Cov(, - / V 0.5 ( V 0.5 ( - fo,..., and some abtal chosen

20 Wocech Gamot 0 m (0,, (0, s needed. Let a vecto U [U,...,U ] consst of ndependent components: U ~Unf(0, and denote: p (m m - (-m (-m m m - ( p 0 m p (3 The fst component of ma be geneated as J (m and subsequent components ae obtaned accodng to the fomula: 0 fo m p J fo m p J 0 (4 whee < a U fo 0 a U fo (a J (5 fo,..., so that E(J (a a. Such a smple pocedue elds a vecto satsfng desed constants snce: m p (m p p p m ( m p m m p m ( m p J E m m p J E 0 0 P( E( P( E( E( and ( V ( V m ( - m ( - m (m m m p m m m m p m m P( P( m m, P( E( E( E(, Cov ( The pocedue depends on the ablt to geneate pseudo-andom numbes U,...,U mtatng ndependent andom vaables havng unfom dstbuton on (0,. Man such geneatos ae wdel avalable ncludng the fast mplementaton of Mesenne-Twste algothm b Matsumoto and shmua (998 mplemented n the R pacage. Ths geneato wll be used n

21 O POOL-ADJACET-VIOLATORS ALGORITHM ou stud. Some sample output of the poposed pocedue wll now be pesented. Fo m [/40, /40, 3/40,...,] and 0 we got a tpcal ealzaton of a bna andom vecto: [ ] Fo m [0.5,...,0.5] and 0 we got a tpcal ealzaton: [ ] Fo m [0.5,...,0.5] and 0.8 we got a tpcal ealzaton: 3 [ ] Fo m [0.5,...,0.5] and -0.8 we got a tpcal ealzaton: 4 [ ] 4. Smulaton esults A smulaton stud was caed out n ode to assess how the bas and mean squae eo of PAVA estmates fo odeed pobabltes depend on the sample sze when vaables ae coelated. Thee smulaton epements wee caed out. In each epement the sequence of n 0,40,...,00 bna vectos was geneated ndependentl h tmes usng the pocedue descbed n pevous secton. All the epements wee caed out usng scpts n R (R Development Coe Team (0. PAVA estmates wee computed usng the gpava functon mplemented n the R sotone pacage (see de Leeuw et al (009 fo a descpton. In the fst epement magnal pobabltes wee set to: m [0.48, 0.49, 0.5, 0.5,0.5] wth 0.0, 0., 0.5, 0.8. In the second epement the wee set to m [0.33, 0.33, 0.35, 0.37,0.37] wth 0.0, 0., 0.5, 0.8. In the thd epement magnal pobabltes amounted to: m 3 m [0.48, 0.49, 0.5, 0.5,0.5]

22 Wocech Gamot wth 0.0, -0., -0.5, Magnal pobabltes wee chosen close to each othe n ode to mae the effects of coectng beached constants b PAVA cleal vsble. The bas and mean squae eo obseved n the fst epement ae shown n fgues and. The bas and mean squae eo obseved n the second epement ae shown n fgues 3 and 4. The bas and mean squae eo obseved n the thd epement ae shown n fgues 5 and Bas( Bas( Bas( Bas( p p p 3 p 4 p Fg.. The bas of PAVA estmates fo m m and 0.0, 0., 0.5, 0.8 In all thee epements the scope of obseved bas depends on the poston of a vaable n the smple ode (. The bas fo estmates p 4 and p 5 of ghtmost pobabltes 4 and 5 tends to be postve whle fo leftmost pobabltes and the bas of estmatos p and p tends to be negatve.

23 O POOL-ADJACET-VIOLATORS ALGORITHM 3 Meanwhle, estmato p 3 of the nnemost pobablt 3 seems to be appomatel unbased. The ntoducton of stong postve coelaton n the fst and second epement seems to educe the bas to some etent. The effect of negatve coelaton assessed n the thd epement s moe comple: estmates of outemost vaables and 5 seem to be unaffected whle the bas of estmates fo and 4 s slghtl educed. Anwa, n all epements and fo all paametes,..., 5 the bas of estmates appaentl tends to zeo when sample sze n nceases. Hence thee s no evdence that the asmptotc unbasedness of PAVA estmates whch was poven b Ae et al (955 unde assumpton of ndependence ceases to hold n the pesence of coelaton MSE(... 5e-04 e-03 e-0 MSE(... 5e-04 e-03 e-0 p p p 3 p 4 p MSE(... 5e-04 e-03 e-0 MSE(... 5e-04 e-03 e Fg.. The MSE of PAVA estmates fo m m and 0.0, 0., 0.5, 0.8

24 4 Wocech Gamot 0 0. Bas( Bas( Bas( Bas( p p p 3 p 4 p Fg. 3. The bas of PAVA estmates fo m m and 0.0, 0., 0.5, 0.8 The mean squae eo of estmates depends on a poston of a vaable n the ode ( as well. In all thee epements the MSE fo estmatos p and p 5 s cleal the hghest of all the fve and onl n the second epement the obseved dffeence between these two s moe ponounced (wth p beng somehow moe accuate than p 5. Anwa, n all epements the MSE of all estmatos p,...,p 5 appaentl tends to zeo wth gowng sample sze whch suggests that consstenc s etaned unde coelaton.

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