Estimators for Finite Population Variance Using Mean and Variance of Auxiliary Variable

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1 Itratoal Jal o Probablt a tattc 5 : - DOI:.59/j.jp.5. tmat Ft Poplato Varac U Ma a Varac o Alar Varabl Ph Mra * R. Kara h Dpartmt o tattc Lcow Urt Lcow Ia Abtract F tmat t poplato arac mato o l alar arabl th m o ma a arac both th Rato-Proct-Drc RPD tp tmat ar propo. Th ralz ca o th tmat la to th cla o tmat ar alo propo. Th ba a ma qar rr M o th propo tmat ar o. Thtcal comparo wth th tratoal tmat ar ppt b a mrcal ampl. B th comparo t how that th propo tmat ar m ct tha th tratoal o. Kw Alar Varabl Tal r pao Ba Ma qar rr M a cc. Itrocto I ampl th alar mato wl at both th ta o lcto a tmato. At th tmato ta alar mato b mlat aro tp o tmat o rt poplato paramtr wth a w o tt cra cc a ar aalabl plt th ltratr. Lt U =... b a t poplato o t wth b th t arabl ta th al th t o U a b th alar arabl ta th al th t o th poplato =.... Lt b th poplato o ma o a b th poplato ma o. Alo lt a r r F a mpl raom ampl o z raw rom U wth th ampl obrato... o a... o lt a b * Crpo ath: r.pmra.a@mal.com Ph Mra Pblh ol at Coprht 5 ctc & Acamc Pblh. All Rht Rr th ampl ma o - al a - al rpctl.. Th t tmat W ow that th t poplato arac t arabl o th whr. From. t atral to t a tmat o w rplac a b thr om tmat. I partclar tmat b a tmat b t th tmat o a ollow w. a t ralz tmat a whr a. a at th alt coto o Tal r pao a bo cto o ch that =.

2 Ph Mra t al.: tmat Ft Poplato Varac U Ma a Varac o Alar Varabl Alo tmat b a tmat b w t aothr tmat o a ollow. a t ralz tmat a.5 whr a a both ar bo cto a rpctl ch that = at th pot = a = at th pot = a both ar at th rlart coto th alt o Tal r pao ha rt two rat wth rpct to a rpctl to b bo.. Ba a Ma qar rr o t tmat a Ba a Ma qar rr o t tmat Lt a F mplct t am that th poplato z lar oh a compar to th ampl z o that t poplato crcto trm ma b. ow.. a. Lt cor th propo tmat.

3 Itratoal Jal o Probablt a tattc 5 : -. Ta pctato o both o. th ba to th r b Ba = U al o th pctato rom. to. w ha Ba.5 ow qar. o both a th ta pctato th ma qar rr o to th rt r o appromato b = M.6 F mmz.6 two ow a th two mal qato atr rtat.6 partall wth rpct to a ar.7.8 ol.7 a.8 a w t th mmz optmm al to b

4 Ph Mra t al.: tmat Ft Poplato Varac U Ma a Varac o Alar Varabl *.9 *. whch wh bttt.6 th mmm al o ma qar rr o th tmat a M m. b Ba a Ma qar rr o t tmat F a b th rt r partal rat o wth rpct to a rpctl at th pot that = a = pa. thr r Tal r abot pot w ha! } * *! whr a ar alra ; a ar th co r partal rat b = = = a * = + h - *= + h - < h <.! * *!

5 Itratoal Jal o Probablt a tattc 5 : - 5. Ta pctato o both o. a al o th pctato rom. to. th ba to th r b Ba Ba. ow qar. o both a th ta pctato th ma qar rr o to th rt r o appromato b M. F mmz. two ow a th two mal qato atr rtat. partall wth rpct to a ar

6 6 Ph Mra t al.: tmat Ft Poplato Varac U Ma a Varac o Alar Varabl.5.6 ol.5 a.6 a w t th mmz optmm al to b *.7 *.8 whch wh bttt. th mmm al o ma qar rr o th tmat a M m.9 c Ba a Ma qar rr o t tmat Lt cor th propo tmat. = =. Ta pctato o both o. th ba p to trm o r b

7 Itratoal Jal o Probablt a tattc 5 : - 7 Ba = U al o th pctato rom. to. w ha Ba =. ow qar. o both a th ta pctato th ma qar rr to th rt r o appromato = = M =. F mmz. two ow a th two mal qato atr rtat. partall wth rpct to a ar.. ol. a. a w t th mmz optmm al to b *.5

8 8 Ph Mra t al.: tmat Ft Poplato Varac U Ma a Varac o Alar Varabl a *.6 whch wh bttt. th mmm al o ma qar rr o th tmat to b M m Ba a Ma qar rr o t tmat F a to b rt co a thr r rat o a to b rt co a thr r rat o a - < h < alo a * = + h - < h < pa *!! *!!.7 at th pot = * = + h at th pot = thr r Tal r w ha * *!!!! = = Ta pctato o both o.8 a al o th pctato rom. to. th ba to th r b.8

9 Itratoal Jal o Probablt a tattc 5 : - 9 Ba.9 ow qar.8 o both a th ta pctato th ma qar rr o appromato b to th rt r o = U al o th pctato rom. to. w ha M = M = F mmz. two ow wth rpct to a a ar. a th two mal qato atr rtat. partall ol. a. a w t th mmz optmm al to b..

10 Ph Mra t al.: tmat Ft Poplato Varac U Ma a Varac o Alar Varabl *. *. whch wh bttt. th mmm al o ma qar rr a M m.5. cc Comparo wth th Tratoal tmat A w ow that th ma qar rr o al cotoal ba tmat arac a M m = M M m = M a m m o poplato.. Comparat t rar thr cc th al cotoal ba tmat ar carr ot wth th hlp o a mrcal lltrato. Cor th ata Cochra 977 al wth Paraltc Polo Ca Placbo rop Paraltc Polo Ca ot oclat rop comptato o rqr al o ha b o a comparo ar ma r a mpl raom ampl o z. F th ata cor w ha = =.58 = 87.6 = = = 7.55 =.5955 =.9688 =

11 Itratoal Jal o Probablt a tattc 5 : - = =.87 8 = 9.59 = U abo al w ha Ma qar rr o al cotoal ba tmat = a M = M = M = M = Hc th prct rlat cc PR o th propo tmat a th al cotoal tmat ar b PR = PR = PR = PR = how that th propo tmat a ar m ct wth hhl cat prct rlat cc th al cotoal ba tmat o th poplato arac. 5. Coclo W ha r w ampl tmat o poplato arac alar mato th m o ma a arac both th ba a ma qar rr qato ar obta. U th qato M o propo tmat ar compar wth th tratoal tmat th a how that th propo tmat ha mallr M tha th tratoal o. ACKOWLDGMT Th ath ar thal to th rr a th t- ch pro alabl to rar mpromt o th papr. RFRC [] Cochra W. C. 977 ampl Tchq r to Joh Wl a o w. [] Da A. K. a Trpath T. P. 978 U o alar mato tmat th t poplato arac. aha Vol. r C 9-8. [] h R. K. Za. M. H. a Rz. A. H 995 tmato o t poplato arac alar mato J. o tattcal t Vol [] h R. K. Za. M. H. a Rz. A. H.996 om tmat o t poplato arac mato o two alar arabl. Mcrolctro Rlab. Vol. 6 o [5] rataa. K. a Jha J. J. H.. 98 A cla o tmat alar mato tmat t poplato arac aha Vol. r C