COMPARISON OF FOUR ESTIMATORS UNDER SAMPLING WITHOUT REPLACEMENT

Transcription

1 JORID 9, Jue, 0. I COPARIO OF FOUR ETIATOR UDER APLIG WITHOUT REPLACEET L. A. afu ad. D. hehu Deartet of atheatcs ad tatstcs, ad A.. alua Deartet of Couter cece, Federal Uverst of Techolog, a E-al:lacoserv@ahoo.co Abstract I soe alcatos, t s cost effcet to sale data ore tha oe stage. I the frst stage, a sle rado sale s draw ad the stratfed accordgl to soe easl easured attrbutes. Ths roect descrbed four estators for the treatet of sales draw wthout relaceet whe equal ad uequal robabltes are cosdered. It also coared ther resultg stadard error uder salg wthout relaceet usg data o dabetc atets ger state for the ears 000, 00, 00 ad 003. The results were obtaed usg a rogra wrtte crosoft Vsual C++ rograg laguage. It was observed that the two-stage salg uder uequal robabltes wthout relaceet s alwas better tha the other three estators cosdered. Kewords: Uequal robablt salg, two-stage salg, hase-hurwtz estator ad horvtz-thoso estator. Itroducto I desgg a stud, t ca be advatageous to sale uts ore tha oe- stage. The crtera for selectg a ut at a gve stage tcall deed o attrbutes observed the revous stages. oe tes of uts a be ore foratve tha others, ad t s better to sale the at a hgher rate. If t costs lttle to detere the attrbutes that are ecessar to classf the uts, t ca be cost effcet to stratf a large sale stage oe ad the stage two to subsale the strata at dfferet rates Chabers ad Dusta, 986. I salg extesve oulatos, rar uts that var sze are ecoutered frequetl. oreover, cosderatos of cost ofte dctate the use of ultstage salg. uose that each ut the oulato ca be dvded to a uber of saller uts, or subuts. A sale of uts has bee selected. If subuts wth a selected ut gve slar results, t sees uecoocal to easure the all Cochra, 977. A coo ractce accordg to Durb 953 ad Thoso 99 s to select ad easure a sale of the subuts a chose ut. Ths techque s called subsalg, sce the ut s ot easured coletel but s tself saled. Aother ae, due to ahalaobs, s two-stage salg, because the sale s take two stes. The frst s to select a sale of uts, ofte called the rar uts, ad the secod s to select a sale of secod stage uts or subuts fro each chose rar ut. ubsalg has a great varet of alcatos, whch go far beod the edate scoe of sale surves. Wheever a rocess volves checal, hscal or bologcal tests that ca be erfored o a sall aout of ateral, t s lkel to be draw as a subsale fro a 3

2 JORID 9, Jue, 0. I larger aout that s tself a sale ardal ad Wrght, 99. Rao et al 990 ad Rao 998 oed that soe cocetrated effort s requred order to obta a good workg kowledge of ultstage salg whe the uts var sze, because the techque s flexble. The uts a be ether wth equal robabltes or wth robabltes roortoal to sze or to soe estate of sze. Varous rules ca be devsed to detere the salg ad subsalg fractos, ad varous ethods of estato are avalable. The advatages of the dfferet ethods deed o the ature of the oulato, o the feld costs, ad o the suleetar data that are at our dsosal. The a obectve of ths research work s to coare four dfferet estators whe data arse fro oe-stage or two-stage desg wth equal or uequal robabltes robabltes roortoal to sze, PP uder salg wthout relaceet. It cosders the stud desg decsos whe salg atets wth health las/health ceters. Te ad source of data The data used ths research work s of secodar te ad was collected fro ger tate str of Health, a. We costructed a sale frae fro all Local Goveret Areas ger tate wth dabetc atets. A coo stud desg health servces research s the ultstage cluster sale Chabers et al, 99 to abstract data fro edcal records. The tcal exale would be the salg of hscas assocated wth those ractces, followed b the salg of atets treated b the saled hscas. The salg la for the stud ust address whch ractces, hscas ad atets to select. The ultate goal s a sale of atets, but the hua subects arovals rocess ad the costs of access to edcal records coel followg the herarch of the servce delver sste to the atet. Joh ad teve 003 cosdered the sle ad coo two-stage roble of salg atets wth hsca ractces to estate a oulato average or total. The referece desg would be to sale ractces roortoal to sze ad the take equal ubers of atets er saled ractce u to the budget costrat. Ths desg zes the desg effect the desg effect secfes the loss or the ga recso of a desg relatve to sle rado salg b sultaeousl zg cluster effects ad the varablt the salg weghts. tud obectve 4 Estato of oulato total I a oe-stage desg such as cluster salg, the varablt of the estator occurs because dfferet sales of rar uts wll gve dfferet values of the estate. Wth two-stage desgs, the estator has varablt eve for a gve selecto of rar uts, because dfferet subsales of secodar uts wll gve rse to dfferet values of the estator. Let deote the uber of rar uts the oulato ad the uber of rar uts the sale. Let be the uber of secodar uts the th rar ut. The total uber of secodar uts the oulato s. Let deote the value of the varable of terest of the th secodar ut the th rar ut. The total of the -values the th rar ut s deoted as Y, that s,. The oulato total s

3 JORID 9, Jue, 0. I uose that the oulato was stratfed to L deedet strata, dexed b s,,, L ad that the ebers of the sth stratu was groued to s rar salg uts PUs, dexed b,,, s ad the th PU cluded s ebers, dexed b,,, s. The corresodg sbols for sale are deoted b the lower case wth the sae subscrts as show table below. Table : bols for two-stage clustered sale data. Poulato Oe-stage Ut Two-stage Ut ale Oe-stage dex Two-stage dex Total Cell Cout Cell Proorto,,,,,,,,,,,, Y Y Y Y Also, table below shows the varaces for the four estators uder salg wthout relaceet. Table : Varaces for the Estated Poulato Total. Varaces for the Estators Equal Probablt Uequal Probablt V Y ewor V Y uwor V Y V Y ewor uwor ethodolog Prar uts selected usg equal robabltes wthout relaceet Whe rar uts are selected b sle rado salg wthout relaceet, a ubased estator of the oulato total s 5

4 JORID 9, Jue, 0. I Y ewor The varace of ths estator s V Y ewor u where u s the fte oulato varace of the rar ut totals, u ad A ubased estate of the varace s Y ewor s u V where s s the sale varace of the rar ut totals, s u u ad 3 4 Prar uts selected usg uequal robabltes wthout relaceet uose that two uts are to be draw fro a stratu. The frst ut s draw wth robabltes, roortoal to soe easure of sze. Let the th ut be selected. If we follow the ost atural ethod, at the secod draw oe of reag uts s selected wth assged robabltes. Hece the total robablt that the th ut wll be selected at ether the frst or the secod draw s + A - where A Also, the robablt that uts ad are the sale s ad. 6 A ubased estator of the oulato total uder salg wthout relaceet wth robabltes roortoal to sze, based o the Horvtz ad Thoso 95, s Y uwor These gve Thus, the varace of Y uwor s gve as: 5 Let Ps deote the robablt of a sale cosstg of secfed uts. The Ps over all sales cotag the th ad th uts, ad Ps over all sales cotag the th ut. Whe we take for, ever Ps for a sale cotag the th ut s couted tes the su, sce there are other values of the sale. Cosequetl,,

5 JORID 9, Jue, 0. I V Y uwor 6 A ubased sale estator of V V Y uwor Provded that oe of the Y uwor s gve as: oulato vashes. 7 Varaces for varous desgs two-stage salg Y ewor Y The varace of V Y ewor where ewor as gve b Okafor 00 s: f f + w Y Y, w uose that we have a ubased estate ubased sale estate of estator of V V where Y ewor Y ewor s: f +, w Uts selected wth equal robabltes wthout relaceet The sale subuts the th ut are chose b sle rado salg. The ubased estator of the oulato total s: Y Y, f ad f of the secod-stage varace 8 9 w of Y ad a fro oe-stage salg. The, a ubased sale w f w, f ad f 0 7

6 JORID 9, Jue, 0. I Uts selected wth uequal robabltes wthout relaceet Let frst salg uts be selected fro the total oulato b robablt roortoal to sze. Let be the robablt of selectg the th frst salg ut. I each selected frst salg, salg uts are selected fro secod b sle rado salg wthout relaceet. The estator of the oulato total s Y uwor Y where + A - where A Also, the robablt The varace of V Y uwor Y uwor where w Y Y ad Its ubased sale estator s V Y uwor where w that uts ad are the sale s as gve b Thoso 99 s: + Y Provded that oe of the Y ad f f + oulato vashes. f f w w 3 Estated stadard errors The results tables 3 6 below for stadard errors were obtaed wth the hel of couter rogra wrtte crosoft Vsual C++ rograg laguage Hubbard, 000. Table 3: tadard Errors for the estated oulato totals usg data ear 000. tadard errors for the estates 8

7 JORID 9, Jue, 0. I Equal Probablt Uequal Probablt Table 4: tadard errors for the estated oulato totals usg data ear 00. tadard Errors for the estates Equal Probablt Uequal Probablt Table 5: tadard errors for the estated oulato totals usg data ear 00. tadard Errors for the estates Equal Probablt Uequal Probablt Table 6: tadard errors for the estated oulato totals usg data ear 003. tadard Errors for the estates Equal Probablt Uequal Probablt error of the estate was vestgated usg Fdgs a ercal ethod wth dfferet estators uder oe-stage salg ad The quatt uder estate s the total two-stage salg. Both tables 3-6 show uber of dabetcs atets b local that the stadard error the case of salg goveret areas ger state for ears uder uequal robablt s less tha the 000, 00, 00 ad 003. The stadard 9

8 JORID 9, Jue, 0. I stadard error of salg uder equal robablt wthout relaceet. Dscusso The data reseted tables 3-6 dcate that substatal reductos the stadard error ca be obtaed through the use of uequal robabltes wthout forfetg a ubased estate of the salg stadard error. Two-stage salg also gves better estato coared wth oe-stage salg. Cocluso ad recoedatos It was observed that two-stage salg uder uequal robabltes wthout relaceet s alwas better. Whe a ubased estator of hgh recso ad a ubased sale estate of ts stadard error are requred, the salg sste elog uequal robabltes at each stage of salg s artcularl arorate. The ultate a of ths stud as stated earler s to coare dfferet estators whe data arse fro oe-stage or two-stage wth equal or uequal robabltes robabltes roortoal to sze, PP uder salg wthout relaceet. Iortatl, two-stage salg uder uequal robabltes robabltes roortoal to sze, PP wthout relaceet was foud to be the ost effcet. Refereces Chabers, R., Dorfa, A. ad Hall, P. 99. Proertes of Estators of the Fte Poulato Dstrbuto Fucto. Boetrka. 79: Chabers, R. ad Dusta, P odel-based Fte Poulato Correcto for the Horvtz-Thoso Estator. Boetrka. 74: Cochra, W.G alg Techques. Thrd edto. ew York: Joh Wle ad os. Durb, J oe Results alg Theor Whe the Uts are elected wth Uequal Probabltes. Joural of the Roal tatstcal ocet. 5: Hase,. ad Hurwtz, W O the Theor of alg Fro Fte Poulatos. Aals of atheatcal tatstcs. 4: Horvtz, D.G. ad Thoso D. J. 95. A Geeralzato of alg Wthout Relaceet Fro a Fte Uverse. Joural of Aerca tatstcal Assocato. 47: Hubbard, J.R Prograg wth C++. ecod Edto. chaus Outles, ew Delh: Tata cgraw- Hll Publshg Coa Lted. Joh, L.A ad teve, L.W alg Patets wth Phsca Practces ad Health las. Joural for Irovg Health Care Delver ad Polc.38: Rao, J., Kovar, J. ad atel, H O Estatg Dstrbuto Fuctos ad Quatles Fro urve Data Usg Auxlar Iforato. Boetrka. 77: Rao, J O the Effcec of alg wth Varous Probabltes ad the electo of Uts wth Relaceet. Boetrka. 83: ardal, C.E. ad Wrght, R.L. 99. Desg-Based ad odel-based Iferece urve alg. cadava Joural of tatstcs. 5: 7-5. Thoso,.K. 99. alg. ew York: Joh Wl ad os. 30