Improved Ratio Estimators for Population Mean Based on Median Using Linear Combination of Population Mean and Median of an Auxiliary Variable

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1 rcan Journal of Opraonal Rsarch : -7 DOI:.59/j.ajor.. Iprov Rao saors for Populaon an as on an Usng Lnar Cobnaon of Populaon an an an of an uxlar arabl Subhash Kuar aav San Sharan shra * lok Kuar Shukla Dparn of ahacs an Sascs Cnr of xcllnc Dr. RL vah Unvrs Fazaba Ina Dparn of Sascs D.- Collg Kanpur Ina bsrac In hs anuscrp h wo ffcn saors of populaon an usng lnar cobnaon of populaon an an an of auxlar varabl hav bn propos. Th xprssons for h bas an an squar rror S hav bn oban up o h frs orr of approxaon. coparson has bn a wh h non xsng saors of populaon an. nurcal su has also bn carr ou o jusf h provn of h propos saors ovr ohr non saors of populaon an of su varabl. Kwors Rao saor Rgrsson saor as an squar rror ffcnc. Inroucon Th propr us of h auxlar varabl whch s hghl posvl or ngavl corrla wh h an varabl unr su nhancs h ffcnc of h saors of h parars unr consraons. L us consr a fn populaon havng snc an nfabl uns. L... b h obsrvaons on h characrsc unr su for h an varabl. L X... b h obsrvaons for h auxlar varabl X. Th auxlar varabl s h varabl abou whch w hav full nforaon. Whn h parars of h auxlar varabl X such as populaon an populaon varanc coffcn of varaon coffcn of kuross coffcn of skwnss an c. ar known hn h us of hs parars provs h ffcnc of h saor. an saors b usng hs parars hav bn propos n h lraur for h provn ovr raonal rao saor. Followng noaons wll b us n hs anuscrp - Sz of h populaon n - Sz of h sapl - Su varabl - an of h su varabl * Corrsponng auhor: san_x@ahoo.co.n San Sharan shra Publsh onln a hp://journal.sapub.org/ajor Coprgh Scnfc & cac Publshng. ll Rghs Rsrv X - uxlar varabl - an of h auxlar varabl ρ - Corrlaon coffcn bwn X an X - Populaon ans x - Sapl ans - vrag of sapl ans of - Sapl an of β - Rgrsson coffcn of on X. - as of h saor. - aranc of h saor S. - an squar rror of h saor S PR p * - Prcnag rlav S p ffcnc of h saor p ovr Followng forula hav bn us n hs papr n C x x X n C S n C

2 Subhash Kuar aav al.: Iprov Rao saors for Populaon an as on an Usng Lnar Cobnaon of Populaon an an an of an uxlar arabl C n f x x X x X n n C x C xx C C n X X n Whr f S Sx X X. s w know ha for an populaon parar h os appropra saor s s corrsponng sasc. So h os appropra saor for populaon an s h sapl an. Th aranc of up o h frs orr of approxaon s S C x C x f. n Th varanc gvn n. s suffcnl larg h a s o ruc hs varanc. Cochran 9 us h auxlar varabl o sa h populaon an of h an varabl an propos h raonal rao saor as Th an squar rror of hs saor up o h frs orr of approxaon s X R. x S R R x R x whr Th raonal rgrsson saor usng auxlar varabl s gvn b Th varanc of h saor lr up o h frs orr of approxaon s lr R X. β X x. lr ρ.5 Subraan an Prabavah propos wo saors of populaon an bas on an of su varabl usng auxlar nforaon as SP SP X X X X Th an squar rrors of hs saors up o h frs orr of approxaon rspcvl ar S R R.8 Whr SP R X. X X an auhors nclung Ssoa an Dwv 98 Srvasava 98 shra an Sahoo 986 Pan an Dub 988 Prasa 989 Sngh an Kakran 99 ohan an Sahoo 995 grawal an Shap 997 Upahaa an Sngh 999 Sngh Sngh an Talor Sngh.al 8 Sngh al. Kalar an Cng 6 Talor an Shara 9 an an Tan aav Pan al. Subraan an.6.7

3 rcan Journal of Opraonal Rsarch : -7 Kuarapanan Solank al. Onka Jlan.al an aav an Kalar c hav us auxlar nforaon for prov saon of populaon an of su varabl.. Propos Iprov Rao saors In hs papr w hav a our frsh ap o xn h saors of Subraan an Prabavah. ovl of prsn work ls n h fac ha whn w nrouc h consans kappas an n h work of Subraan an Prabavah w hav bn abl o sk h nw saors whch ar foun or ffcn n h procss of opal sarch. W hav rv h xprssons for h bass an h an squar rrors for h propos saors. Th an squar rrors of h propos saors conan h unknown consans kappas. Ths consans ar oban b nzng h an squar rrors of h propos saors usng ho of axa an na. Whn h opu valus of hs saors whch ar oban b xnng h prvous work ar pu n h xprssons of h an squar rror h an squar rrors ar nz as copar o Subraan an Prabavah an ohr xsng saors lang o sr opu sarch of ffcn saors for populaon an of h gvn populaon. ova b Subraan an Prabavah an Prasa 989 w hav propos wo prov rao saors of populaon an as X. X X. X To su h larg sapl proprs of propos saors an l us fn as an such ha an. xprssng saor n rs of s w hav X X X On subracng boh h ss of abov quaon w hav up o h frs orr of approxaon... Takng xpcaon on boh ss of abov quaon w g bas of as

4 Subhash Kuar aav al.: Iprov Rao saors for Populaon an as on an Usng Lnar Cobnaon of Populaon an an an of an uxlar arabl ] [ ] [ as. Th an squar rror of up o h frs orr of approxaon s ] [ S ] [ On splfng w hav }] } [ S. To fn h nu valu of h S w us h axa na ho. In hs ho w ffrna S wh rspc o an qua o zro an solvng hs quaon for w fn h opu valu of whch nzs h valu of S. Th opu valu of s Whr Pung h opu valu of n quaon. w g h nu S of as n S.5 Slarl h bas an h an squar rror of up o h frs orr of approxaon rspcvl ar ] [ as.6 }] } [ S.7 Whch s nu for Whr Th nu S of s gvn b

5 rcan Journal of Opraonal Rsarch : ffcnc Conons Fro quaon. an.5 w hav S n.8 S S > Th propos saor n.5 s br han h rao saor n. f Fro quaon.5 an.5 w hav f f S >. n S R S > or [ R x R x ] >. S lr S > or [ ρ ] >. Fro quaon.8 an.5 w hav S SP S > f o: Slar conons ar for h saor.. urcal xapl [ R R ] >. To ou h horcal fnngs w hav consr h populaon fro Sngh an Chauhar 986 pag 77. Th aa sascs s as follows Tabl. Populaon Parars for sapl sz n X R. 999 R x ρ. 9 x 56. Tabl. Populaon Parars for sapl sz 5 n X R. 999 R x ρ. 9 x

6 6 Subhash Kuar aav al.: Iprov Rao saors for Populaon an as on an Usng Lnar Cobnaon of Populaon an an an of an uxlar arabl Tabl. an squar rrors an prcnag rlav ffcncs of ffrn saors saor For sapl sz n For sapl sz n 5 S PR S PR R lr SP SP Propos Propos Rsuls an Conclusons Fro h horcal vlopns of h anuscrp an h rsuls n Tabl- w nfr ha h propos saors an of populaon an hav h nu an squar rrors han h non saors h sapl an rao saor rgrsson saor an h saors of Subraan an Prabavah. Thus h propos saors shoul b prfrr for h saon of populaon an. RFRCS [] grawal.c. an Shap.. 997: Hrarchc prcv rao-bas an prouc-bas saors an hr ffcnc. Journal of ppl Sascs 97-. [] Cochran W.G. 9: Th saon of h ls of h cral xprns b saplng for h Rao of gran o oal prouc Jour. gr. Sc [] Jlan.I. aqbool S. an r S.. : of Rao saors of Populaon an Usng Lnar Cobnaon of Co-ffcn of Skwnss an Quarl Dvaon. Inrnaonal Journal of orn ahacal Scncs [] Kalar C. an Cng H. : Rao saors n spl rano saplng ppl ahacs an Copuaon [5] Kalar C. an Cng H. 6: n provn n sang h populaon an b usng h corrlaon co-ffcn Hacp Journal of ahacs an Sascs olu 5-9. [6] shra G. an Sahoo L.. 986: Class of ransfor rao an prouc saors n sapl survs. Journ. Sa. Rsarch [7] ohan S. an Sahoo J. 995: no on provng h rao ho of saon hrough lnar ransforaon usng cran known populaon parars. Sankha [8] urh.. 967: Saplng Thor an hos Sascal.Publshng Soc Calcua. [9] Onka C saon of populaon an n possraf saplng usng known valu of so populaon parars. Sascs n Transon-w Srs :65-78 [] Prasa. 989: So prov rao p saors of populaon an an rao n fn populaon sapl survs Councaons n Sascs: Thor an hos [] Pan.. an Dub. 988: of prouc saor usng coffcn of varaon of auxlar vara. ssa Sa. Rvw [] Pan H aav SK Shukla K n prov gnral class of saors sang populaon an usng auxlar nforaon. Inrnaonal Journal of Sascs an Sss 6:-7 [] Robson D.S. 957: pplcaonf ulvara polkas o h hor of unbas rao p saon. J.r.sa. ssoc.5 -. [] Solank RS Sngh HP Rahour n alrnav saor for sang h fn populaon an usng auxlar nforaon n sapl survs. Inrnaonal Scholarl Rsarch work: Probabl an Sascs :- [5] Sngh H.P. an Kakran.S. 99: of rao saor usng coffcn of kuross of an auxlar characrunpublsh anuscrp [6] Sngh G.. : On h provn of prouc ho of saon n sapl survs. JISS [7] Srvasava S.K. 98: Prcv saon of fn populaon an usng prouc saor. rka 9-99.

7 rcan Journal of Opraonal Rsarch : -7 7 [8] Sngh D. an Chauhar F.S 986. Thor an analss of sapl surv sgns w g Inrnaonal Publshr. [9] Sngh H.P. an Talor R. : Us of known corrlaon co-ffcn n sang h fn populaon ans Sascs n Transon [] Sngh H.P. Talor R. Talor R. an Kakran.S. : n prov saor of populaon an usng powr ransforaon Journal of h Inan Soc of grculural Sascs [] Ssoa..S. an Dwv.K. 98: of rao saor usng co-ffcn of varaon of auxlar varabl Journal of h Inan Soc of grculural Sascs -8. [] Subraan J. an Kuarapanan G. : saon of Populaon an Usng Co-ffcn of araon an an of an uxlar arabl. Inrnaonal Journal of Probabl an Sascs : -8. [] Subraan J. an Prabavah G.. an as of Rao saors wh Lnar Cobnaons of Populaon an an an of an uxlar arabl Journal of Rlabl an Sascal Sus 7 -. [] Upahaa L.. an Sngh H.P. 999: Us of ransfor auxlar varabl n sang h fn populaon an orcal Journal [5] aav SK ffcn saors for populaon varanc usng auxlar nforaon. Global Journal of ahacal Scncs: Thor an Praccal :69-76 [6] aav S.K. an Kalar C. : Iprov class of rao an prouc saors ppl ahacs an Copuaon [7] an Z. an Tan. : Rao ho o h an saon usng co-ffcn of skwnss of auxlar varabl ICIC Par II CCIS 6 pp..