Estimation of Population Variance Utilizing Auxiliary Information

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1 Iteratioal Joural of Statistics ad Systems ISSN Volume 1, Number (017), pp Research Idia Publicatios Estimatio of Populatio Variace Utilizig Auxiliary Iformatio Sheela Misra 1, * Dipika ad Dharmedra Kumar Yadav 3 Departmet of Statistics, Uiversity of Luckow, Luckow-6007, Idia. (* Correspodig author) Abstract I this article, a estimatio procedure for the populatio variace utilizig auxiliary iformatio ad kow coefficiet of variatio is proposed. The Bias ad mea square error of proposed estimator are foud up to first order of approximatio. A comparative study with the usual ubiased estimator ad usual ratio estimator for populatio variace has bee made. Numerical study is also give at the ed of the article to support the theoretical fidigs. Keywords: Bias, Coefficiet of Variatio, Efficiecy, Mea Square Error, Simple Radom Samplig. 1. INTRODUCTION Sometimes, additioal iformatio o some other variable highly correlated with the characteristic uder study is available. This additioal iformatio is kow as auxiliary or acillary or priori iformatio ad the character o which additioal iformatio is provided kow as auxiliary or acillary character. This auxiliary iformatio may be kow i advace from the past data, pilot survey or from the experiece of the observer. Auxiliary iformatio is used to improve the efficiecy of the estimator. I statistics it is proved that use of auxiliary iformatio i probability samplig cosiderably reduces the variace of the estimator of populatio parameter. Such as i may agricultural surveys for estimatig total productio of

2 30 Sheela Misra, Dipika ad Dharmedra Kumar Yadav ay crop, area of crop cultivatio are used as auxiliary iformatio Here our proposed estimator uses the auxiliary iformatio available o variable uder study. Let the study variable y ad auxiliary variable takig the values Yi ad Xi respectively for the ith (i=1,,..., N) uit of the populatio of size N. Such that N N i=1, Y = 1 Y N i=1 i, X = 1 X N i μ rs = 1 N (X i X ) r (Y i Y ) s C y = σ y Y, β y = μ 0, γ 1y = μ 03 3/ μ 0 μ 0 C x = σ x X, β x = μ 0, γ 1x = μ 30 3/ N σ y = 1 N (Y i Y ) i=1 μ 0 N μ 0, σ x = 1 N (X i X ) Let yi ad xi are the observatio of sample values of study ad auxiliary variables respectively. For estimatig populatio variace the proposed estimator is s yα = s y (x C x α s ) x Where α is the characterizig scalar chose suitably. i=1 (1). BIAS AND MEAN SQUARE ERROR OF PROPOSED ESTIMATOR For the sake of simplicity we are assumig that the populatio size N is large as compared to sample size so that fiite populatio correctio is igored. Let, So that x = X (1 + e 0 ) s y = σ y (1 + e 1 ) s x = σ x (1 + e ) E(e 0 ) = E(e 1 ) = E(e ) = 0

3 Estimatio of Populatio Variace Utilizig Auxiliary Iformatio 305 E(e 0 ) = C x E(e 1 ) = γ y + E(e ) = γ x + E(e 0 e 1 ) = λc x Also, we have E(e 0 e ) = γ 1xC x E(e 1 e ) = δ 1 δ = μ σ x σ, λ = μ 1 y σ x σ, γ x = μ 0 3, γ y = μ 0 3 y From (1), writig s yα i terms of ei s = σ y (1 + e 1 ) (1 αe + μ 0 μ 0 α s yα = σ y (1 + e 1 ) [X (1 + e 0 ) C x σ x (1 + e ) ] +.. ) α(α + 1) e + αe 0 α e 0 e + α(α 1)e 0 (s yα σ y ) = σ y (αe 0 + e 1 αe + α(α 1)e 0 + Takig α e 0 e αe 1 e ) () α(α + 1) e + αe 0 e 1 Expectatio o both sides of (), we get bias up to I st order of approximatio Bias(s yα ) = σ y [αe(e 0 ) + E(e 1 ) αe(e ) + α(α 1)E(e 0 ) + α(α + 1) E(e ) + αe(e 0 e 1 ) α E(e 0 e ) αe(e 1 e )]

4 306 Sheela Misra, Dipika ad Dharmedra Kumar Yadav = σ y α (C x + γ x γ 1x C x + ) σ y (C x λc x γ x + δ ) (3) Now for mea square error squarig ad takig expectatio o both sides of () we get MSE of s yα as MSE(s yα ) = σ y [k E(e 0 ) + E(e 1 ) + k E(e ) + ke(e 0 e 1 ) k E(e 0 e ) ke(e 1 e )] = σ y (γ y + ) + σ y α (C x + γ x + γ 1x C x ) + σ y α(λc x δ + ) () The optimum value of α which miimizes the mea square error of s yα i () is give by α 0 = λc y δ+1 C x γ 1x C x +γ x + (5) The miimum value of mea square error of proposed estimator s yα for α 0 is give by MSE(s yα ) = σ y mi (γ y + ) σ y ( (λc y δ + 1) C x γ 1x C x + γ x + ) (6) 3. THEORETICAL EFFICIENCY COMPARISON (a) Efficiecy compariso of proposed estimator to usual ubiased estimator for populatio variace MSE(s yα ) mi MSE(s y ) < 0 σ y (γ y + ) σ y ( (λc y δ + 1) C x γ 1x C x + γ x + ) σ y (γ y + ) < 0 δ λc x > 1 (7) (b) Efficiecy compariso of proposed estimator with ratio estimator of populatio variace MSE(s yα ) mi MSE(s R ) < 0

5 Estimatio of Populatio Variace Utilizig Auxiliary Iformatio 307 Where, σ y (γ y + ) σ y ( (λc y δ + 1) C x γ 1x C x + γ x + ) σ y [(γ y + ) + (γ x + ) (δ 1)] < 0C > (AB) 1 (8) A = γ 1x C x γ x C x B = γ x δ + 1 C = λc x δ + 1 Proposed estimator is better that usual ubiased estimator of populatio variace ad ratio estimator if the data follows the coditios defied i (7) ad (8) respectively.. ILLUSTRATION For the umerical compariso betwee proposed estimator to sample variace ad ratio estimator, we cosider the data give i Cochra (1977, page 181) dealig with paralytic polio cases placebo (y) group, computatio of required values have bee doe ad we have, Y =.588, X =.9, σ y = 9.890, σ x =.639, C x = β 1y =.318, β y =.337, β 1x = 3.9, β x = γ 1y = 1.5, γ y = 1.337, γ 1x = 1.981, γ x = 3.391, δ = 3.85, λ = 1.11 Table 1: MSE s of Estimator Estimators Mea Square Error MSE(s y ) MSE(s R ) MSE(s yα ) mi 6.757

6 308 Sheela Misra, Dipika ad Dharmedra Kumar Yadav The percet relative efficiecy (PRE) of the proposed estimator over the usual ubiased estimator for populatio variace is 1% ad the percet relative efficiecy (PRE) of the proposed estimator over ratio estimator of populatio variace is 153%. 5. CONCLUDING REMARKS (a) From (6), for the optimum value of α, the miimum mea square error attaied by estimator s yα is give by MSE(s yα ) mi = σ y (γ y + ) σ y ( (λc y δ + 1) C x γ 1x C x + γ x + ) (b) The coditios i which proposed estimator will perform better tha usual ubiased estimator ad ratio estimator are derived i (7) ad (8). (c) From umerical illustratio it is observed that proposed estimator is 133% efficiet from usual ubiased estimator for populatio Variace ad 153% efficiet from Ratio estimator. REFERENCES [1] Agrawal, M. C. ad Pada, K. B. (1999): A predictive justificatio for variace estimatio usig auxiliary iformatio. Jour. Id. Soc. Ag. Stat.,5(), [] Biradar, R. S. ad Sigh, H. P. (1998): Predictive estimatio of fiite populatio variace. Cal. Statist. Assoc. Bull., 8, [3] Blad, J. M. ad Altma, D. G. (1986): Statistical method for assessig agreemet betwee two methods of cliical measuremet, Lace, 1(876), [] Cochra, W.G.(1963), Samplig Techiques, Secod Editio, Wiley Easter Private Limited, New Delhi. [5] Chaudhury, A. (1978): O estimatig the variace of a fiite populatio. Metrika, 5, [6] Das, A. K. ad Tripathi, T. P. (1977): Admissible estimators for quadratic forms i fiite populatios. Bull. Iter. Stat. Ist., 7(),

7 Estimatio of Populatio Variace Utilizig Auxiliary Iformatio 309 [7] Das, A. K. ad Tripathi, T. P. (1978), Use of auxiliary iformatio i estimatig the fiite populatio variace, Sakhya, c,, [8] Gupta, S. ad Shabbir, J. (008).Variace estimatio i simple radom samplig usig auxiliary iformatio. Hacettepe Joural of Mathematics ad Statistics, 37, [9] Isaki, C.T.(1983), Variace estimatio usig Auxiliary Iformatio. Jour. Amer. Statist. Asssoct., 78, [10] Kadilar, C. ad Cigi, H. (006a). Improvemet i variace estimatio usig auxiliary iformatio. Joural of Mathematics ad Statistics, 35(1), [11] Kadilar, C. ad Cigi, H. (006 b).ratio estimators for populatio variace i simple ad stratified samplig. Applied Mathematics ad Computatio, 173, [1] Liu, T. P (197): A geeralized ubiased estimator for the variace of a fiite populatio, Sakhya, 36, C, 3 3. [13] Mukhopadhyay, P. (1978): Estimatig a fiite populatio variace uder a super populatio model, Metrika, 5, [1] Mukhopadhyay, P. (198): Optimum Strategies for estimatig the variace of a fiite populatio uder a super populatio model, Metrika, 9, [15] Padmwar, V. R. ad MukhopadhyaY, P. (1981): Estimatio of symmetric fuctios of a fiite populatio, Metrika, 31, [16] Sukhatme P.V., Sukhatme B.V., Sukhatme, S. ad Ashok, C. (198), Samplig Theory of Surveys with Applicatios, Iowa State Uiversity Press, Ams. [17] Swai, A. K. P. C. ad Mishra, G. (199): Estimatio of populatio variace uder uequal probability samplig, Sakhya, Ser B, 56, [18] Tripathi, T. P., Sigh, H. P. ad Upadhyaya, L. N. (00): A geeral method of estimatio ad its applicatio to the estimatio of coefficiet of variatio, Statistics i Trasitio, 5(6), [19] Wakimoto, K. (1971): Stratified radom samplig (I): Estimatio of Populatio variace, A. Ist. Stat. Math., 3, [0] Wolter, K. M. (1985). Itroductio to variace estimatio. New York, NY: Spriger- Verlag.

8 310 Sheela Misra, Dipika ad Dharmedra Kumar Yadav