On Some New Allocation Schemes in Stratified Random Sampling under Non-Response

Transcription

1 Maoj Kr. Chaudhary, V. K. gh, Rajesh gh Deartmet of tatstcs, Baaras Hdu Uversty Varaas-5, IDIA Floret maradache Deartmet of Mathematcs, Uversty of ew Mexco, Gallu, UA O ome ew Allocato chemes tratfed Radom amlg uder o-resose Publshed : Rajesh gh, F. maradache (Edtors) UDIE I AMPLIG ECHIQUE AD IME ERIE AALYI Z Publshg, Columbus, UA, IB

2 Abstract hs chater resets the detaled dscusso o the effect of o-resose o the estmator of oulato mea a frequetly used desg, amely, stratfed radom samlg. I ths chater, our am s to dscuss the exstg allocato schemes resece of o-resose ad to suggest some ew allocato schemes utlzg the owledge of resose ad o-resose rates of dfferet strata. he effects of roosed schemes o the samlg varace of the estmator have bee dscussed ad comared wth the usual allocato schemes, amely, roortoal allocato ad eyma allocato resece of o-resose. he emrcal study has also bee carred out suort of the results. Keywords: tratfed radom samlg, Allocato schemes, o-resose, Mea squares, Emrcal tudy.. Itroducto uhatme (95) has show that by effectvely usg the otmum allocato stratfed samlg, estmates of the strata varaces obtaed a revous survey or a secally laed lot survey based eve o samles of moderate samle sze would be adequate for creasg the recso of the estmator. Evas (95) has also cosdered the roblem of allocato based o estmates of strata varaces obtaed earler survey. Accordg to lterature of samlg theory, varous efforts have bee made to reduce the error whch arses because of tag a art of the oulato,.e., samlg error. Besdes the samlg error there are also several o-samlg errors whch tae lace from tme to tme due to a umber of factors such as faulty method of selecto ad estmato, comlete coverage, dfferece tervewers, lac of roer suervso, etc. Icomleteess or o-resose the form of absece, cesorg or groug s a troublg ssue of may data sets.

3 I choosg the samle szes from the dfferet strata stratfed radom samlg oe ca select t such a way that t s ether exclusvely roortoal to the strata szes or roortoal to strata szes alog wth the varato the strata uder roortoal allocato or eyma allocato resectvely. If o-resose s heret the etre oulato ad so are all the strata, obvously t would be qute mossble to adot eyma allocato because the the owledge of stratum varablty wll ot be avalable, rather the owledge of resose rate of dfferet strata mght be easly avalable or mght be easly estmated from the samle selected from each stratum. hus, t s qute reasoable to utlze the resose rate (or o-resose rate) whle allocatg samles to stratum stead of eyma allocato resece of o-resose error. I the reset chater, we have roosed some ew allocato schemes selectg the samles from dfferet strata based o resose (o-resose) rates of the strata resece of o-resose. e have comared them wth eyma ad roortoal allocatos. he results have bee show wth a umercal examle.. amlg trategy ad Estmato Procedure I the study of o-resose, accordg to oe determstc resose model, t s geerally assumed that the oulato s dchotomzed two strata; a resose stratum cosderg of all uts for whch measuremets would be obtaed f the uts haeed to fall the samle ad a o-resose stratum of uts for whch o measuremet would be obtaed. However, ths dvso to two strata s, of course, a oversmlfcato of the roblem. he theory volved HH techque, s as gve below: Let us cosder a samle of sze s draw from a fte oulato of sze. Let uts the samle resoded ad uts dd ot resod, so that +.

4 he uts may be regarded as a samle from the resose class ad uts as a samle from the o-resose class belogg to the oulato. Let us assume that ad be the umber of uts the resose stratum ad o-resose stratum resectvely the oulato. Obvously, ad are ot ow but ther ubased estmates ca be obtaed from the samle as ˆ / ; ˆ /. Let m be the sze of the sub-samle from o-resodets to be tervewed. Hase ad Hurwtz (96) roosed a estmator to estmate the oulato mea X of the study varable X as HH x + x m, (.) whch s ubased for X, whereas x ad x m are samle meas based o samles of szes ad m resectvely for the study varable X. he varace of HH s gve by L V ( HH ) +, (.) where L, m, ad are the mea squares of etre grou ad oresose grou resectvely the oulato. Let us cosder a oulato cosstg of uts dvded to strata. Let the sze of th stratum s, (,,..., ) ad we decde to select a samle of sze from the etre oulato such a way that uts are selected from the th stratum. hus, we have.

5 Let the o-resose occurs each stratum. he usg Hase ad Hurwtz rocedure we select a samle of sze m uts out of o-resodet uts the th stratum wth the hel of smle radom samlg wthout relacemet (ROR) such that method. L m, L ad the formato are observed o all the m uts by tervew he Hase-Hurwtz estmator of oulato mea X for the th stratum wll be x + x m, (,,..., ) (.) where x ad x m are the samle meas based o resodet uts ad m oresodet uts resectvely the th stratum. Obvously s a ubased estmator of X. Combg the estmators over all strata we get the estmator of oulato mea X, gve by st (.) where. Obvously, we have E[ ] X. (.5) st he varace of st s gve by [ ] V st + ( L ) (.6)

6 where, ad are the mea squares of etre grou ad o-resose grou resectvely the th stratum. for It s easy to see that uder roortoal allocato (PA), that s, whe,,...,, V [ ] s obtaed as st [ st ] PA V + ( L ), (.7) whereas uder the eyma allocato (A), wth equal to (,,..., ), t s [ st ] A V + ( L ). (.8) It s mortat to meto here that the last terms the exressos (.7) ad (.8) arse due to o-resose the oulato. Further, resece of o-resose the oulato, eyma allocato may or may ot be effcet tha the roortoal allocato, a stuato whch s qute cotrary to the usual case whe oulato s free from o-resose. hs ca be uderstood from the followg: e have V [ ] [ ] ( ) ( ) + w st V PA st A w L (.9) w.

7 hole the frst term the above exresso s ecessarly ostve, the secod term may be egatve ad greater tha the frst term magtude deedg uo the w sg ad magtude of the term for all. hus, resece of o-resose the stratfed oulato, eyma allocato does ot always guaratee a better result as t s case whe the oulato s free from o-resose error.. ome ew Allocato chemes It s a well ow fact that case the stratfed oulato does ot have oresose error ad strata mea squares, (,,..., ), are ow, t s always advsable to refer eyma allocato scheme as comared to roortoal allocato scheme order to crease the recso of the estmator. But, f the oulato s affected by o-resose, eyma allocato s ot always a better roosto. hs has bee hghlghted uder the secto above. Moreover, case o-resose s reset strata, owledge o strata mea squares,, are mossble to collect, rather drect estmates of ad may be had from the samle. Uder these crcumstaces, t s, therefore, ractcally dffcult to adot eyma allocato f o-resose s heret the oulato. However, roortoal allocato does ot demad the owledge of strata mea squares ad rests oly uo the strata szes, hece t s well alcable eve the resece of o-resose. As dscussed the secto, ubased estmates of resose ad o-resose rates the oulato are readly avalable ad hece t seems qute reasoable to th for develog allocato schemes whch volve the owledge of oulato resose (o-resose) rates each stratum. If such allocato schemes yeld récsed estmates as comared to roortoal allocato, these would be advsable to adot stead of eyma allocato due to the reasos metoed above. I ths secto, we have, therefore, roosed some ew allocato schemes whch utlze the owledge of resose (o-resose) rates suboulatos. hle some of the roosed schemes do ot utlze the owledge of, some others are roosed 5

8 based o the owledge of just order to mae a comarso of them wth eyma allocato uder the resece of o-resose. I addto to the assumtos of roortoal ad eyma allocatos, we have further assume t logcal to allocate larger samle from a stratum havg larger umber of resodets ad vce-versa whe roosg the ew schemes of allocatos. cheme-[oa ()]: Let us assume that larger sze samle s selected from a larger sze stratum ad wth larger resose rate, that s, for,,...,. he we have K where K s a costat. he value of K wll be K. hus we have. (.) Puttg ths value of exresso (.6), we get V ( L ) [ st ] + (.) 6

9 7 cheme-[oa ()]: Let us assume that. he, we have (.) ad hece the exresso (.6) becomes [ ] + st L V ) (. (.) cheme-[oa ()]: Let us select larger sze samle from a larger sze stratum but smaller sze samle f the o-resose rate s hgh. hat s,. he (.5) ad the exresso of [ ] st V reduces to [ ] ( ) { } + st L V. (.6)

10 8 cheme-[oa ()]: Let, the. (.7) he corresodg exresso of [ ] st V s [ ] ( ) + st L V. (.8) cheme-5[oa (5)]: Let, the. (.9) he exresso (.6) gves [ ] ( ) + st L V 5. (.)

11 cheme-6[oa (6)]: If, the, we have. (.) st I ths case, [ ] V becomes V [ ] st 6 ( L ) +. (.) Remar : It s to be metoed here that f resose rate assumes same value all the strata, that s (say), the schemes, ad 5 reduces to roortoal allocato, whle the schemes, ad 6 reduces to eyma allocato. he corresodg V exressos, [ st ] r reduce to V [ st ] A, (,,5 ). V V r are the smlar to [ st ] PA ad [ st ] r Remar : Although the theoretcal comarso of exressos of V [ st ] r ad V [ st ] r, (,,6) V r wth [ st ] PA ad [ st ] A, ( r,,6), ( r,,5 ) V resectvely s requred order to uderstad the sutablty of the roosed schemes, but such comarsos do ot yeld exlct solutos geeral. he sutablty of a scheme does deed uo the arametrc values of the oulato. e have, therefore, llustrated the results wth the hel of some emrcal data. 9

12 . Emrcal tudy I order to vestgate the effcecy of the estmator st uder roosed allocato schemes, based o resose (o-resose) rates, we have cosdered here a emrcal data set. e have tae the data avalable ardal et. al. (99) gve Aedx B. he data refer to 8 mucaltes wede, varyg cosderably sze ad other characterstcs. he oulato cosstg of the 8 mucaltes s referred to as the MU8 oulato. For the urose of llustrato, we have radomly dvded the 8 mucaltes to four strata cosstg of 7, 7, 97 ad mucaltes. he 985 oulato ( thousads) has bee cosdered as the study varable, X. O the bass of the data, the followg values of arameters were obtaed: able : Partculars of the Data ( 8) tratum () ze ( ) tratum Mea ( X ) tratum Mea quare ( ) Mea quare of the oresose Grou ( ) e have tae samle sze, 6. 5

13 ables dects the values of samle szes, (,,, ) ad values of V [ ] uder PA, A ad roosed schemes OA() to OA(6) for dfferet selectos of the (,,, ) values of L ad. st 5

14 tratum oresose Rate ( ) (Percet) able amle zes ad Varace of st uder Dfferet Allocato chemes L.,.5,.5,.5 for,,, resectvely) ( amle ze ( ) ad [ ] V uder PA A OA() OA() OA() OA() OA(5) OA(6) V [ st ] [ ] V st [ ] V st [ ] V st st [ ] V st [ ] V st [ ] V st V [ ] st

15 5. Cocludg Remars I the reset chater, our am was to accommodate the o-resose error heret the stratfed oulato durg the estmato rocedure ad hece to suggest some ew allocato schemes whch utlze the owledge of resose (o-resose) rates of strata. As dscussed dfferet sub-sectos, eyma allocato may sometmes roduce less récsed estmates of oulato mea comarso to roortoal allocato f o-resose s reset the oulato. Moreover, eyma allocato s sometmes mractcal such stuato, sce the ether the owledge of (,,, ), the mea squares of the strata, wll be avalable, or these could be estmated easly from the samle. I cotrast to ths, what mght be easly ow or could be estmated from the samle are resose (o-resose) rates of dfferet strata. It was, therefore, thought to roose some ew allocato schemes deedg uo resose (o-resose) rates. A loo of able reveals that most of the stuatos (uder dfferet combatos of ad L ), allocato schemes OA (), OA () ad OA (5), deedg solely uo the owledge of ad (or ), roduce more récsed estmates as comared to PA. Further, as for as a comaratve study of schemes OA (), OA () ad OA (5) s cocered, o doubt, all these schemes are more or less smlar terms of ther effcecy. hus, addto to the owledge of strata szes,, the owledge of resose (o-resose) rates, (or ), whle allocatg samle to dfferet strata; certaly adds to the recso of the estmate. It s also evdet from the table that the addtoal formato o the mea squares of strata certaly adds to the recso of the estmate, but ths cotrbuto s ot very much sgfcat comarso to A. cheme OA () s throughout worse tha ay other scheme. 5

16 Refereces Evas,. D. (95) : O stratfcato ad otmum allocato. Jour. of he Amer. tat. Assoc., 6, 95-. Hase, M. H. ad Hurwtz,.. (96) : he roblem of o-resose samle urveys. Jour. of he Amer. tat. Assoc.,, ardal, C. E., wesso, B. ad retma, J. (99) : Model Asssted urvey amlg. rger-verlag, ew Yor, Ic. uhatme, P. V. (95) : Cotrbutos to the theory of the reresetatve Method. Jour. of the Royal tat. oc.,,