Multivariate Ratio Estimation With Known Population Proportion Of Two Auxiliary Characters For Finite Population

Transcription

1 Multvarate Rato Estmaton Wth Knon Populaton Proporton Of To Auxlar haracters For Fnte Populaton *Raesh Sngh, *Sachn Mal, **A. A. Adeara, ***Florentn Smarandache *Department of Statstcs, Banaras Hndu Unverst,Varanas-5, Inda ** Department of Statstcs, Unverst of Ilorn, Ilorn, Kara State, Ngera *** har of Department of Mathematcs, Unverst of Ne Mexco, Gallup, USA Abstract In the present stud, e propose estmators based on geometrc and harmonc mean for estmatng populaton mean usng nformaton on to auxlar attrbutes n smple random samplng. We have shon that, hen e have mult-auxlar attrbutes, estmators based on geometrc mean and harmonc mean are less based than Oln (958), Na and Gupta (996) and Sngh (967) tpe- estmator under certan condtons. Hoever, the MSE of Oln( 958) estmator and geometrc and harmonc estmators are same up to the frst order of proxmaton. Ke ords: Smple random samplng, auxlar attrbute, pont b-seral correlaton, harmonc mean, geometrc mean.. Introducton Pror noledge about populaton mean along th coeffcent of varaton of the populaton of an auxlar varable s non to be ver useful partcularl hen the rato, product and regresson estmators are used for estmaton of populaton mean of a varable of nterest. There exst stuatons hen nformaton s avalable n the form of the attrbute hch s hghl correlated th. For example ma be the use of drugs and ma be gender. Usng the nformaton of pont bseral correlaton beteen the stud varable and the auxlar attrbute Na and Gupta (996), Shabbr and Gupta (6), Ab-Alfatah et al. () and Sngh et al. (7, 8) have suggested mproved estmators for estmatng unnon populaton mean Y.

2 Usng nformaton on mult-auxlar varables postvel correlated th the stud varable, Oln (958) suggested a multvarate rato estmator of the populaton mean Y. In ths per, e have suggested some estmators usng nformaton on mult-auxlar attrbutes. Follong Oln (958), e defne an estmator as r P (.) here () s are eghts such that = () P ' s are the proporton of the auxlar attrbute and assumed to be non and () r, p s the sample mean of the stud varable Y and p s the proporton of auxlar attrbutes P based on a smple random sample of sze n dran thout replacement from a populaton of sze N. Follong Na and Gupta (996) and Sngh et al. (7), e propose another estmator t s as t s r P (.) To alternatve estmators based on geometrc mean and harmonc mean are suggested as r P (.3) and r P (.4) such that

3 These estmators are based on the assumptons that the auxlar attrbutes are postvel correlated th Y. Let P and P and (=,, ; =,, ) be the ph correlaton coeffcent beteen be the correlaton coeffcent beteen Y and P. S N N P, So Y Y N S S and, N Y P In the same a and are defned. ' Further, let,,..., and (,,...;,,..., ) pp. BIAS AND MSE OF THE ESTIMATORS To obtan the bas and MSE s of the estmators, up to frst order of proxmaton, let e Y and Y e p P P such that E(e ) (,,,..., ).. Expressng equaton (.) n terms of e s, e have e e Y 3 = Y e e e e e e e e (.) Subtractng Y from both the sdes of equaton (.) and then tang expectaton of both sdes, e get the bas of the estmator up to the frst order of proxmaton as f Y B (.) Subtractng Y from both the sdes of equaton (.) squarng and then tang expectaton of both sdes, e get the bas of the estmator up to the frst order of proxmaton as MSE f Y I (.3)

4 To obtan the bas and MSE of (.3) n term of e s, as to the frst order of proxmaton, e express equaton Y e e = Y e e e e (e ee ) (.4) Subtractng Y from both sdes of equaton (.4) and then tang expectaton of both sdes, e get the bas of the estmator t up to the frst order of proxmaton, as f Y B (.5) Subtractng Y from both the sdes of equaton (.4) squarng and then tang expectaton of both sdes, e get the bas of the estmator up to the frst order of proxmaton as MSE f Y I (.6) No expressng equaton (.4) n terms of e s, e have e e Y 3 e e e Y e Y e e ee e (.7) Subtractng Y from both sdes of equaton (.7) and then tang expectaton of both sdes, e get the bas of the estmator up to the frst order of proxmaton ll be f Y B (.8)

5 Subtractng Y from both the sdes of equaton (.7) squarng and then tang expectaton of both sdes, e get the bas of the estmator up to the frst order of proxmaton as f Y MSE (.9) We see that MSE s of these estmators are same and the bases are dfferent. In general, MSE = MSE = MSE. (.) 3. omparson of bases The bases ma be ether postve or negatve. So, for comparson, e have compared the absolute bases of the estmates hen these are more effcent than the sample mean. The bas of the estmator of geometrc mean s smaller than that of arthmetc mean B > B (3.) Squarng and smplfng (3.), e observe that 3 (3.) Thus above nequalt s true hen both the factors are ether postve or negatve. The frst factor of (3.) 3 s postve, hen 3 ' (3.3)

6 In the same a, t can be shon that the second factor of (3.) s also postve hen ' (3.4) When both the factors of (3.) s negatve, the sgn of nequaltes of (3.3) and (3.4) reversed. Also comparng the square of the bases of geometrc and harmonc estmator, e fnd that geometrc estmator s more based than harmonc estmator. Hence e ma conclude that under the stuatons here arthmetc, geometrc and harmonc estmator are more effcent than sample mean and the relaton (3.4) or ' 3 s satsfed, the bases of the estmates satsf the relaton B > B > B Usuall the eghts of s are so chosen so as to mnmze the MSE of an estmator subect to the condton. 4. Emprcal Stud Data : (Source: Sngh and haudhar (986), P. 77). The populaton conssts of 34 heat farms n 34 vllages n certan regon of Inda. The varables are defned as: = area under heat crop (n acres) durng 974. p = proporton of farms under heat crop hch have more than 5 acres land durng 97. and

7 p = proporton of farms under heat crop hch have more than acres land durng 973. For ths data, e have N=34, Y =99.4, P =.6765, P =.7353, S =564.6, S =.549, S =.535, pb =.599, pb =.559, =.75. Bases and MSE s of dfferent estmators under comparson, based on the above data are gven n Table 4.. TABLE 4. : Bas and MSE of dfferent estmators Estmators Auxlar attrbutes Bas MSE none Rato Rato P p P p P P Oln ( ) P and P Suggested P and P Suggested P and P t s P p P p P and P

8 5. oncluson From Table 4. e observe that the MSE s of Oln (958) tpe estmator, estmator based on harmonc and geometrc mean are same. Sngh (967) tpe estmator t s performs orse. Hoever, the bas of the rato-tpe estmator based on harmonc mean s least. Hence, e ma conclude that hen more than one auxlar attrbutes are used for estmatng the populaton parameter, t s better to use harmonc mean. References. Abd-Elfattah, A.M. El-Sherpen, E.A. Mohamed, S.M. Abdou, O. F. (): Improvement n estmatng the populaton mean n smple random samplng usng nformaton on auxlar attrbute. Appl. Math. and ompt. do:.6/.amc.9..4 Na, V.D and Gupta, P.. (996): A note on estmaton of mean th non populaton proporton of an auxlar character. Jour. Ind. Soc. Agr. Stat., 48(), Oln, I. (958): Multvarate rato estmaton for fnte populaton. Bometrca, Shabbr, J. and Gupta, S. (6): A ne estmator of populaton mean n stratfed samplng, ommun. Stat. Theo. Meth. 35: 9 Sngh, D. and haudhar, F. S. (986) : Theor and Analss of Sample Surve Desgns John Wle and Sons, NeYor. Sngh, R., auhan, P., Saan, N. and Smarandache,F. (7): Auxlar nformaton and a pror values n constructon of mproved estmators, Renassance Hgh press, USA. Sngh, R., hauhan, P., Saan, N. and Smarandache,F. (8): Rato estmators n smple random samplng usng nformaton on auxlar attrbute. Pa. J. Stat. Oper. Res.,4,, Sngh, M.P. (967): Multvarate product method of estmaton for fnte populatons. J. Indan Soc. Agr. Statst., 9, -.