r Laka Joural o Appl tatstcs Vol 5-3 stmato o Populato Varac Us a Gralz Doubl ampl stmator Push Msra * a R. Kara h Dpartmt o tatstcs D.A.V.P.G. Coll Dhrau- 8 Uttarakha Ia. Dpartmt o tatstcs Luckow Uvrst Luckow- 67 Uttar Prash Ia. * Corrspo Author: r.pmsra.av@mal.com Rcv: 7 Auust 3 / Rvs: 5 Ma ; Auust / Accpt: ovmbr IApptat-L 3 ABRAC For th stmato o t populato varac a ralz oubl sampl stmator s propos. h bas a ma squar rror M o th propos stmator ar ou. hortcal comparso wth th tratoal stmator s carr out a t s show that th propos stmator s mor ct tha th tratoal o. Kwors: Aular varabl alor s srs paso Bas Ma quar rror.. Itroucto I sampl thor aular ormato s wl us at th stas o slcto a stmato at th slcto sta th aular ormato s us b s varous sampl schms a at th stmato sta t s us ormulat varous tps o stmators o rt populato paramtrs wth a vw o tt cras cc. stmators lk rato prouct rc rrsso a th classs o rato a prouct tp stmators or populato paramtrs mal populato ma a varac ar stu b ma authors a ar avalabl th ltratur. o ct som rrcs ths cott o ma s stmato procurs a thr proprts b Das a rpath 978 Lu 97 a rvastava a Jhajj 98. But wh paramtrs o o or mor aular varabls ar ot avalabl avac th th altratv s to us oubl sampl or two phas sampl tchqu whr w rst tak a prlmar lar sampl o IAL IB-39-987
Push Msra a R. Kara h sz call rst phas sampl o whch ol th aular varabl s obsrv a th rom tak a sub-sampl o sz call sco phas sampl o whch both th varabls ar obsrv. I such stuatos th rt stmators kow as oubl sampl rato prouct rc a rrsso stmators wr vlop. For mor tals rar othr oubl sampl stmato procurs s ukhatm t. al. 98 Murth 967 a Cochra 977. hs prst papr too cotrbuts to ths ara. Lt a lar prlmar sampl o sz s raw rom a populato o sz a th a subsampl o sz rom s raw b us smpl raom sampl wthout rplacmt schm or both th phass. At rst phas sampl o sz ol th aular varabl s obsrv a at th sco phas sampl o sz both th stu varabl a th aular varabl ar obsrv. Lt b th sampl mas o bas o sco phas sampl o sz a b th sampl ma o rst phas sampl valus o th aular charactr. Lt b th populato corrlato coct btw a a r rs s whr ar th populato valus o rspctvl or th th =... ut o th populato. For stmat t populato varac a ralz oubl sampl stmator s propos as. whr sats th valt cotos o alor s srs paso s a bou ucto o such that. rst orr partal rtal coct o wth rspct to at s ut that s.3 IB-39-987 IAL
stmato o Populato Varac Us a Gralz Doubl ampl stmator sco orr partal rtal coct o wth rspct to at s zro that s. v = -.5 or a b th rst orr partal rvatvs o wth rspct to a rspctvl at th pot a v = -.6 or a.. om Partcular Mmbrs Blo to th Propos stmator om partcular mmbrs blo to ths propos ralz oubl sampl stmator ar: a b t ˆ t ˆ k c t ˆ k 3 k ˆ t k k ˆ k3 k 3 t 5 k whr k k k a k 3 ar th charactrz scalars to b chos sutabl a ˆ. IAL IB-39-987 3
Push Msra a R. Kara h IB-39-987 IAL 3. Bas a Ma quar rror o Propos stmator h propos ralz oubl sampl stmator as quato. s ˆ whr ˆ. Lt = = = a = or Z z whr z = & Z or Z z or ˆ whr z ˆ a Z. For smplct w assum that th populato sz s lar ouh as compar to th sampl sz so that th t populato corrcto trms ma b or. o w hav = = = = 3. 3.
stmato o Populato Varac Us a Gralz Doubl ampl stmator IAL IB-39-987 5 3 3 3.3 ow smlar to h 98 a apat a ahoo 6 pa sa t th thr orr alor s srs about th pot a ot that w hav t = {! * * * 3! 3 O mplo th cotos rom. to.6 w hav t =! * * * 3! 3 3. whr a ar alra rom.3 to.6 sco orr partal rvatvs a ar v b a * h * h * h or < h <.
Push Msra a R. Kara h 6 IB-39-987 IAL ow us 3.. a tak appromato w hav!!. 3.5 ak pctato o both ss o 3.5 a or trms Murth 967 th bas up to trms o orr s v b. Us valus o th pctatos v 3. to 3.3 w hav
stmato o Populato Varac Us a Gralz Doubl ampl stmator IAL IB-39-987 7 Bas 3.6 ow squar 3.5 o both th ss a th tak pctato th ma squar rror to th rst r o appromato s v b!. Us valus o th pctatos v 3. to 3.3 w hav M = 3 3. 3.7 O mplo th coto th ma squar rror o bcoms M =
Push Msra a R. Kara h =. 3.8 h optmum valu o mmz th ma squar rror o * s v b 3.9 whch wh substtut 3.8 vs th mmum valu o ma squar rror o as M m. 3. As w kow that th ma squar rror o usual covtoal ubas stmator s o populato varac s a rom 3. th ma squar rror o th propos stmator b coms out to show that th ma squar rror o th propos stmator s lss tha that o th usual covtoal ubas stmator s o populato varac. mprcal tu. For compar cc o th propos stmator lt us cosr th ata v Cochra 977 al wth Paraltc Polo cass Placbo roup a Paraltc Polo cass ot oculat roup. W hav calculat th rqur valus o a a comparso s ma. rs For 3 a 5 sa w hav = 9.889 = 7.86588 7 =.9688 = 93.675 3 Ma quar rror o usual covtoal ubas stmator s 9.5336697 a Ma quar rror o th propos stmator s 8.399538. h prct rlatv cc PR o th propos stmator ovr th usual covtoal ubas stmator s 3.63568 whch shows that th propos stmator s mor ct tha th usual covtoal ubas stmator. 8 IB-39-987 IAL
stmato o Populato Varac Us a Gralz Doubl ampl stmator Ackowlmt h authors ar thakul to th rrs a tor or thr valuabl commts a sustos rar mprovmt o th papr. Rrcs. Cochra W. G. 977 ampl chqus 3 r to Joh Wll a os w ork.. Das A.K. a rpath.p. 978 Us o aular ormato stmat th t populato varac. akha C 39-8. 3. Lu.P. 97 A ral ubas stmator or th varac o a t populato. akha C 36 3-3.. Murth M.. 967 ampl hor a Mthos st to tatstcal Publsh oct Calcutta Ia. 5. apat. C. a ahoo L.. 6 A altratv class o stmators oubl sampl Bullt o th Malasa Mathmatcal ccs oct 9 89-9. 6. h R. Kara. 98 Gralz oubl sampl stmators or th rato a prouct o populato paramtrs Joural o Ia tatstcal Assocato 39-9. 7. rvastava.k. a Jhajj H.. 98 A class o stmators us aular ormato or stmat t populato varac. akha C 87-96. 8. ukhatm P.V. ukhatm B.V. ukhatm. a Asok C. 98 ampl thor o survs wth applcatos 3 r to Ams Iowa UA a Ia oct o Arcultural tatstcs w Dlh Ia. IAL IB-39-987 9
Push Msra a R. Kara h IB-39-987 IAL