SOME IMPROVED ESTIMATORS IN MULTIPHASE SAMPLING ABSTRACT KEYWORDS

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1 Pak. J. Statist. 010, Vol SOME IMPROVED ESTIMATORS IN MULTIPHASE SAMPLING Muhammad Hanif 1, Muhammad Qaiser Shahbaz, and Zahoor Ahmad 3 1 Lahore University of Management Sciences, Lahore, Pakistan hanif@lums.edu.pk Department of Mathematics, COMSATS Institute of IT, Lahore, Pakistan qshahbaz@gmail.com 3 Department of Statistics, University of Gujrat, Gujrat, Pakistan zahoorstat@yahoo.com ABSTRACT Some improved estimators of population mean has been proposed in two phase and multiphase sampling using information on two and several auxiliary variables. The minimum variance of the proposed estimators has been obtained. Comparison of the proposed estimators has been done with some available estimators of two phase sampling that utilizes information of two and several auxiliary variables. KEYWORDS Regression estimators, correlation coefficients, two phase sampling, optimum variance. 1 INTRODUCTION The use of auxiliary information has always been a source of improvement in estimation of certain population characteristics. The auxiliary variables that have string relationship with the estimand variable always improve the precision of the estimates resulting in smaller standard error of the estimate. The use of auxiliary variables in survey sampling has very old history. Several regression type estimators are available in literature that uses the auxiliary information to increase the precision of the estimates. The historical use of the auxiliary variables can be found in the regression estimator given by Hansen et al 1953 as: ȳ lr = ȳ + β X x; 010 Pakistan Journal of Statistics 195

2 196 where β is the regression coefficient of Y on X. The variance of classical regression estimator is: Var ȳ lr = θs y 1 ρ yx ; 1.1 where θ = n 1 N 1 and ρ yx is the correlation coefficient between X and Y. The use of several auxiliary variable in the context of regression estimator has been discussed by Ahmad 008. The multiple regression estimator has the general form: ȳ lr = ȳ + β X x; where β is the regression coefficient of Y on X. The variance of classical regression estimator is: Var ȳ lr = θs y 1 ρ yx ; 1. where ρ y.x is the squared multiple correlation coefficient between Y and the combined effect of all the auxiliary variables. The regression estimator has been effectively used in two phase and multiphase sampling. The estimator of mean in two phase sampling has been discussed by Hansen et al 1953 and is given as: ȳ lr = ȳ + β x 1 x ; where x 1 and x are first phase and second phase mean of auxiliary variable X based upon the samples of sizes n 1 and n respectively; ȳ is mean of Y for the second phase sample of size n. The variance of regression estimator for two phase sampling is given as: Var ȳ lr = S y { θ 1 ρ yx + θ1 ρ yx} ; 1.3 where θ h = n 1 h N 1 and n h is sample size at hth phase. A slightly modified regression type estimator has been proposed by Sahoo et al 1993 by using information of two auxiliary variables. The proposed estimator is: ȳ ssm = ȳ + β 1 x 1 x + β W w 1 ; The variance of estimator given by Sahoo et al 1993 is: Some Improved Estimators in Multiphase Sampling Var ȳ ssm = S y { θ 1 ρ yx + θ1 ρ yx ρ yw} ; 1.4 where ρ yw is squared correlation coefficient between Y and W. Kiregyera 1984 has proposed the following regression-in-regression estimator: ȳ K = ȳ + β yx [ x 1 x + β wx W w 1 ];

3 Hanif, Shahbaz and Ahmad 197 The variance of ȳ K is: Var ȳ K = Sy [ θ 1 ρ yx + θ1 ρ yx + ρ yxρ ] wx ρ yx ρ yw ρ wx ; 1.5 Roy 003 has given the following regression type estimator: ȳ R = ȳ + α[ x 1 + β W w 1 { x + γ W w }]; The variance of estimator given by Roy 003 is: Var ȳ R = Sy [ θ 1 ρ y.wx + θ1 ρ ] yx.w 1 ρ yw ; 1.6 where ρ y.wẋ is the squared multiple correlation between Y and combined effects of X and W, ρ yx.w is the squared partial correlation between Y and X keeping W at a constant level and ρ yw is the squared correlation between Y and W. Another regression type estimator has been proposed by Samiuddin and Hanif 007. The estimator is given as: ȳ sh = ȳ + α x 1 x + β W w 1 + γ W w ; 1.7 Samiuddin and Hanif 007 has also proposed the following ratio type estimators using two auxiliary variables: α1 α X X z α3 ȳ sh = ȳ ; 1.8 x 1 x z Samiuddin and Hanif 007 has shown that the variance of 1.7 and 1.8 is same as the variance of Roy s estimator given in 1.6. Mukerjee et al 1987 has also proposed a regression type estimator using two auxiliary variables. The estimator is: The variance of ȳ M is given as: ȳ M = ȳ + β yx x 1 x + β yw w 1 w ; Var ȳ M = Sy [ θ θ θ 1 ρ yw + ρ ] yx ρ yx ρ yw ρ wx ; 1.9 Chand 1975 has proposed various ratio type estimators. One of the estimator proposed by Chand 1975 is given as: ȳ C = ȳ x w 1 x 1 W ; The variance of above estimator is of complex form. In this paper we have proposed a modified regression type estimator using information on several auxiliary variables.

4 198 Some Improved Estimators in Multiphase Sampling THE PPROPSED ESTIMATOR-1 We have proposed the following estimator of population mean in two phase sampling: ȳ sh1 = ȳ + α 1 x 1 x + α w 1 w + α 3 W w 1 ;.1 where α 1,α and α 3 are the unknowns to be determined by minimizing the variance of.1. Writing ȳ = Ȳ +ē y, x 1 = X +ē x1, x = X +ē x, w 1 = W +ē w1 and w = W +ē w we have: ȳ sh1 Ȳ = ē y + α 1 ē x1 ē x + α ē w1 ē w α 3 ē w1 ; Squaring and applying the expectation we have: Var ȳ sh1 = θ S y + α 1 θ θ 1 S x + α θ θ 1 S w + α 3θ 1 S w α 1 θ θ 1 S xy α θ θ 1 S wy α 3 θ 1 S wy +α 1 α θ θ 1 S wx. The optimum values of α 1,α and α 3, that minimizes. are: α 1 = S y ρ xy ρ xw ρ yw S x 1 ρ wx α = S y ρ wy ρ xw ρ xy S w 1 ρ wx α 3 = S yρ yw S w = β yw ; = β yx.w = β yw.x Using these values in.; and simplifying; the variance of.1 is given as: Var ȳ sh1 = Sy [ θ 1 ρ y.wx + θ1 ρ ] yx.w 1 ρ yw ;.3 The variance given in.3 is same as the variance of the regression type estimator given by Roy 003, but the formation of Roy s estimator is relatively complex as compared with the estimator proposed in.1. Further, by using the relation 1 ρ y.wx = 1 ρ yw 1 ρ yx.w the variance of ȳ sh1 can also be written as: Var ȳ sh1 = Sy [ θ 1 ρ y.wx + θ1 ρ y.wx ρ ] yw ;.4 The proposed estimator for multiphase sampling can be directly written as: ȳ sh1 = ȳ h + α 1 x m x h + α w m w h + α 3 W w m ;

5 Hanif, Shahbaz and Ahmad 199 with variance given as: Var ȳ sh1 = Sy [ θh 1 ρ y.wx + θm ρ ] yx.w 1 ρ yw ;.5 A consistent estimator of population mean can be easily written from.1 as: ȳ sh1 = ȳ + b yx.w x 1 x + b yw.x w 1 w + b yw W w 1 ;.6 with estimator of variance given as: var ȳ sh1 = s y [ θ 1 r y.wx + θ1 r yx.w 1 r yw ] ;.7 The confidence interval for population mean can be easily constructed by using.6 and.7. 3 THE PROPOSED ESTIMATOR- We propose the following estimator of population mean by using information of several auxiliary variables: ȳ sh = ȳ + α x 1 x + β W w 1 ; 3.1 where x 1 is vector of first phase mean of p auxiliary variables X, x is vector of second phase mean of p auxiliary variables X, w 1 is vector of first phase mean of q auxiliary variables W and W is the vector of population means for variable W s. The vectors α and β are the vectors of unknown parameters whose values are to be determined so that the variance of 3.1 is minimum. Now, using ȳ = Ȳ + ē y, x 1 = X + ē x1, x = X + ē x, w 1 = W + ē w1 in 3.1 we have: ȳ sh Ȳ = ē y + α ē x1 ē x β ē w1 ; Squaring and applying expectation, the variance of ȳ sh is given as: Var ȳ sh = θ S y + θ θ 1 α S x α + θ 1 β S w β θ θ 1 α s xy θ 1 β s wy. 3. where S x is the covariance matrix of x, S w is the covariance matrix of w, s xy is vector of covariances between Y and x and s wy is vector of covariances between Y and w. Partially differentiating 3. w.r.t.α and β and setting the derivatives to zero we obtain α = S 1 x s xy and β = S 1 w s wy. Further, by using the value of α and β in 3., the variance of 3.1 is: Var ȳ sh = Sy [ θ 1 ρ y.x + θ1 ρ y.x ρ ] y.w ; 3.3

6 00 Some Improved Estimators in Multiphase Sampling where ρ y.x is the multiple correlation coefficient between Y and x and ρ y.w is the multiple correlation coefficient between Y and w. The estimator ȳ sh for multiphase sampling can be analogously written from 3.1 as: with variance given as: ȳ sh = ȳ h + α x m x h + β W w m ; Var ȳ sh = Sy [ θh 1 ρ y.x + θm ρ y.x ρ ] y.w. 3.4 The consistent estimator and estimator of variance can be straight away written from 3.1 and 3.3. In the following section we have given the comparison of the proposed estimator with the available estimators of two phase sampling. 4 COMPARISON WITH AVAILABLE Ahmad 008 has proposed various estimators for two phase and multiphase sampling using information on several auxiliary variables. One of the estimator proposed by Ahmad 008 with two auxiliary variables is given as: The variance of above estimator is given as: t 1 = ȳ + α 1 x 1 x + α w 1 w ; Var t 1 = Sy [ θ 1 ρ y.wx + θ1 ρ ] yx.w. 4.1 Also the variance of the estimator proposed by Sahoo et al 1993 is given in 1.4. Now comparing.3 with 4.1 we have: Var t 1 Var ȳ sh1 = θ 1 S yρ yx.wρ yw 4. Now 4. is always positive. Hence the proposed estimator ȳ sh1 always outperform the estimator t 1 proposed by Ahmad 008. Now, to compare the proposed estimator ȳ sh1 with the estimator proposed by Sahoo et al 1993, we consider 1.4 and.4. From these two equations we have: Var ȳ ssm Var ȳ sh1 = θ θ 1 S y ρ y.wx ρ yx. 4.3

7 Hanif, Shahbaz and Ahmad 01 The quantity given in 4.3 is always positive and hence the proposed estimator ȳ sh1 will always have smaller variance than the estimator given by Sahoo et al Further, Ahmad 008 has proposed the following regression estimator for two phase sampling using multiple auxiliary variables: The variance of t is given as: Now, comparing 3.3 with 4.4 we have: t = ȳ + α x 1 x. Var t = Sy [ θ 1 ρ y.x + θ1 ρ ] y.x. 4.4 Var t Var ȳ sh = θ 1 S yρ y.w. 4.5 The quantity given in 4.5 is always positive which clearly shows that the new estimator ȳ sh outperform the estimator proposed by Ahmad 008. Acknowledgement The authors are thankful to the referee for valuable comments which helped a lot in improving the quality of the paper. REFERENCES 1. Ahmad, Z. 008 Generalized Multivariate Ratio and Regression Estimators for Multi-phase Sampling, Unpublished PhD thesis, Submitted at National College of Business Administration and Economics, Lahore, Pakistan.. Chand, L Some Ratio-type estimators based on two or more auxiliary variables, Unpublished PhD thesis, Submitted to Iowa State University, Iowa. 3. Hansen, M. H., Hurwitz, W. N. and Madow, W. G Sample Survey Methods and Theory, Vol II, John Wiley, New York. 4. Kiregyera, B Regression-type estimator using two auxiliary variables and model of double sampling from finite populations. Metrika. 31, Mukerjee, R., Rao, T.J. and Vijayan, K Regression type estimators using multiple auxiliary information. Austral. J. Statist. 93,

8 0 Some Improved Estimators in Multiphase Sampling 6. Roy, D. C. 003 A regression type estimator in two phase sampling using two auxiliary variables, Pak. J. Stat., 193, Samiuddin, M. and Hanif, M. 007 Estimation of population mean in single and two phase sampling with or without additional information, Pak. J. Stat., 3, Sahoo, J., Sahoo, L. N. and Mohanty, S A regression approach to estimation in two phase sampling using two auxiliary variables. Current Sciences, 651, Samiuddin, M. and Hanif, M. 007 Estimation of population mean in single and two phase sampling with or without additional information, Pak. J. Stat., 3,