A General Family of Estimators for Estimating Population Variance Using Known Value of Some Population Parameter(s)

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1 Rajesh Sigh, Pakaj Chauha, Nirmala Sawa School of Statistics, DAVV, Idore (M.P.), Idia Floreti Smaradache Uiversity of New Meico, USA A Geeral Family of Estimators for Estimatig Populatio Variace Usig Kow Value of Some Populatio Parameter(s) Published i: Rajesh Sigh, Pakaj Chauha, Nirmala Sawa, Floreti Smaradache (Editors) AUXILIARY INFORMATION AND A PRIORI VALUES IN CONSTRUCTION OF IMPROVED ESTIMATORS Reaissace High Press, A Arbor, USA, 007 (ISBN-0): (ISBN-3): pp

2 Abstract A geeral family of estimators for estimatig the populatio variace of the variable uder study, which make use of kow value of certai populatio parameter(s), is proposed. Some well kow estimators have bee show as particular member of this family. It has bee show that the suggested estimator is better tha the usual ubiased estimator, Isaki s (983) ratio estimator, Upadhyaya ad Sigh s (999) estimator ad Kadilar ad Cigi (006). A empirical study is carried out to illustrate the performace of the costructed estimator over others. Keywords: Auiliary iformatio, variace estimator, bias, mea squared error.. Itroductio I maufacturig idustries ad pharmaceutical laboratories sometimes researchers are iterested i the variatio of their produce or yields (Ahmed et.al. (003)). Let (U = U, U,..., U N ) deote a populatio of N uits from which a simple radom sample without replacemet (SRSWOR) of size is draw. Further let y ad deote the study ad the auiliary variables respectively. 65

3 Let N Y = y i ad y = ( yi Y) N S deotes respectively the ukow N populatio mea ad populatio variace of the study character y. Assume that populatio size N is very large so that the fiite populatio correctio term is igored. It is established fact that i most of the survey situatios, auiliary iformatio is available (or may be made to be available divertig some of the resources) i oe form or the other. If used itelligibly, this iformatio may yield estimators better tha those i which o auiliary iformatio is used. Assume that a simple radom sample of size is draw without replacemet. The usual ubiased estimator of S y is s y = ( yi y) (.) y = y i is the sample mea of y. Whe the populatio mea square = ( i X) proposed a ratio estimator for S y as N S is kow, Isaki (983) N = S s t (.) s i is a ubiased estimator of = ( ) S. 66

4 Several authors have used prior value of certai populatio parameter(s) to fid more precise estimates. The use of prior value of coefficiet of kurtosis i estimatig the populatio variace of study character y was first made by Sigh et. al. (973). Kadilar ad Cigi (006) proposed modified ratio estimators for the populatio variace usig differet combiatios of kow values of coefficiet of skewess ad coefficiet of variatio. I this paper, uder SRSWOR, we have suggested a geeral family of estimators for estimatig the populatio variace S y. The epressios of bias ad mea-squared error (MSE), up to the first order of approimatio, have bee obtaied. Some well kow estimators have bee show as particular member of this family.. The suggested family of estimators Motivated by Khoshevisa et. al. (007), we propose followig ratio-type estimators for the populatio variace as ( as b) t = s y (.) [ α(as b) + ( α)(as b)] (a 0), b are either real umbers or the fuctio of the kow parameters of the auiliary variable such as coefficiet of variatio C() ad coefficiet of kurtosis ( β ()). The followig scheme presets some of the importat kow estimators of the populatio variace, which ca be obtaied by suitable choice of costats α, a ad b: 67

5 Table. : Some members of the proposed family of the estimators t Estimator Values of α a b t 0 = s y S s t = Isaki (983) 0 estimator t = [S C ] Kadilar s C C ad Cigi (006) estimator t t3 = [S β s β() ()] = [Sβ () sβ() C t5 = [SC β()] s C β () s y t 6 = [S + β s + β () C ()] ] β () β () C C β () - β () Upadhyaya ad Sigh (999) The MSE of proposed estimator t ca be foud by usig the firs degree approimatio i the Taylor series method defied by MSE (t) d d (.) 68

6 h = [ h(a, b) a S y, S h(a, b) b S y, S ] V(s y ) = Cov(s,s y ) Cov(s V(s y,s ) ). Here h(a,b) = h( s y, s ) = t. Accordig to this defiitio, we obtai d for the proposed estimator, t, as follows: αasy d = [ - ] as + b MSE of the proposed estimator t usig (.) is give by as y αas y MSE(t) V(s y ) Cov(s y,s ) V(s ) as b as b α + (.3) V(s y) = λsy[ β(y) ] V(s ) = λsy[ β() ] Cov(,s ) = λsys (h ) (.) μ0 μ0 λ =, β (y) =, β () =, μ μ 0 0 h μ =, μ0μ0 μ N r s rs = (yi Y) (i X) N Usig (.), MSE of t ca be writte as, (r, s) beig o egative itegers. { β (y) αθ(h ) + α θ ( β () ) } MSE(t) λsy (.5) as θ =. as b The MSE equatio of estimators listed i Table. ca be writte as- 69

7 { β (y) αθ (h ) + α θ ( β () ) } MSE(ti ) λsy i i, i =,3,, 5,6 (.6) S θ =, S C S θ 3 =, S β () Sβ () θ =, Sβ () C SC θ 5 =, S C β () S θ 6 =. S + β () Miimizatio of (.5) with respect to α yields its optimum value as C αopt α = = (.7) θ (h ) C =. { β () } By substitutig α opt i place of α i (.5) we get the resultig miimum variace of t as [ β (y) { β () } ] mi.mse(t) = λsy (.8) 3. Efficiecy comparisos correctio) of Up to the first order of approimatio, variace (igorig fiite populatio t o = s y ad t is give by [ β (y) ] Var() = λsy (3.) [{ β (y) } + { β () }( C) ] MSE(t) = λs y (3.) From (.6), (.8), (3.), ad (3.), we have { β () } C 0 Var( ) mi.mse(t) = λsy > (3.3) { β () }( θ C ) 0 MSE(ti ) mi.mse(t) = λsy i >, i =,,3,, 5,6 (3.) provided C θ. i 70

8 Thus it follows from (3.3) ad (3.) that the suggested estimator uder optimum coditio is (i) always better the s y, (ii) better tha Isaki s (983) estimator t ecept whe C = i which both are equally efficiet, ad (iii) Kadilar ad Cigi (006) estimators t i (i =,3,,5) ecept whe C = θi (i =,3,,5) i which t ad t i (i =,3,,5) are equally efficiet.. Empirical study We use data i Kadilar ad Cigi (00) to compare efficiecies betwee the traditioal ad proposed estimators i the simple radom samplig. I Table., we observe the statistics about the populatio. Table.: Data statistics of the populatio for the simple radom samplig N = 06, = 0, ρ = 0. 8, C y =. 8, C =. 0, Y = 5. 37, X = 3. 76, S y = 6.5, S = 9. 89, β () 5. 7, β (y) 80. 3, λ = 0. 05, θ = = = The percet relative efficiecies of the estimators s y, t i (i =,3,,5,6 ) ad mi.mse(t) with respect to s y have bee computed ad preseted i Table. below. 7

9 Table.: Relative efficiecies (%) of s, t i (i =,3,,5,6 ) ad mi.mse(t) with respect y to s y. Estimator PRE (., s ) t 0 = s y 00 t t t t t t mi.mse(t).39 y 5. Coclusio From theoretical discussio i sectio 3 ad results of the umerical eample, we ifer that the proposed estimator t uder optimum coditio performs better tha usual estimator s y, Isaki s (983) estimator t, Kadilar ad Cigi s (006) estimators (t, t 3, t, t 5 ) ad Upadhyaya ad Sigh s (999) estimator t 6. Refereces Ahmed, M.S., Dayyeh, W.A. ad Hurairah, A.A.O. (003): Some estimators for fiite populatio variace uder two-phase samplig. Statistics i Trasitio, 6,,

10 Isaki, C.T. (983): Variace estimatio usig auiliary iformatio. Jour. Amer. Stat. Assoc., 78, 7-3. Kadilar, C. ad Cigi, H. (006): Ratio estimators for the populatio variace i simple ad stratified radom samplig. Applied Mathematics ad Computatio 73 (006) Sigh, J., Padey, B.N. ad Hirao, K. (973): O the utilizatio of a kow coefficiet of kurtosis i the estimatio procedure of variace. A. Ist. Statist. Math., 5, Upadhyaya, L.N. ad Sigh, H. P. (999): A estimator for populatio variace that utilizes the kurtosis of a auiliary variable i sample surveys. Vikram Mathematical Joural, 9,

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