Improved Estimation of Current Population Mean Over Two Occasions

This paper addresses the role of two auxiliary variables on both the occasions to improve the precision of estimates at current (second) occasion in two occasion successive sampling .Utilizing the readily available information on two auxiliary variables on both the occasions and information on the study variable from the previous occasion, an efficient estimation procedure of population mean on current (second) occasion has been envisaged. It is to be mentioned that out of the two auxiliary variables one auxiliary variable is positively correlated with the study variable while other is negatively correlated. Behavior of the proposed estimator has been studied and compared with the sample mean estimator, when there is no matching from the previous occasion and traditional successive sampling estimator, which is a linear combination of the means of the matched and unmatched portions of the sample at the current (second) occasion. Optimal replacement policy is also discussed. In addition, we support the theoretical results with the aid of numerical examples.


Introduction
It is well recognized fact that the use of auxiliary information in estimating the parameters such as population mean or total improve the precision of estimates. If the survey is repetitive in nature, past values may also be used as auxiliary information to improve the precision of the current estimates. Jessen (1942) was the first who introduced the procedure of utilizing the information obtained on first occasion in improving the estimates of the current occasion. Estimation on more than two occasions is due to Patterson (1950). The theory of Patterson (1950) has been extended by Eckler (1955), Rao and Graham (1964), Cochran (1977), Sen 2 ISSN 2424-6271 IASSL (1971,1973), Sukhatme et al. (1984) and Singh et al. (1992), Singh et al. (2014) among others. In many situations of practical importance, information on an auxiliary variable may be readily available on the first as well as on the second occasion, for instance, see Singh andVishwakarma (2007a, b, 2009), Singh and Kumar (2010), Kumar (2012) and Singh et al. (2013) .The aim of this paper is to develop a procedure of utilizing the information on two auxiliary variables readily available on both the occasions. We have assumed a situation, where one auxiliary variable is positively correlated and the other auxiliary variable is negatively correlated with the study variable on the first and second occasions. In this situation, we have made an effort to propose an estimator of the population mean on current occasion using information on two auxiliary variates on both occasions in two occasion successive sampling besides the information on the study variable on the previous occasion. Theoretical properties of the suggested estimator have been discussed and numerical illustrations are given. Numerical results show the dominance of the suggested estimator over the sample mean estimator and the natural successive sampling estimator when no auxiliary information is used. Suitable recommendations have been made on the basis of numerical findings.
The motivation in defining the estimator u d is taken from Bahl and Tuteja (1991).
The second estimator is based on the sample of size     n m common to both the occasions given by The estimator u d may be used to estimate the population mean on each occasion, while the estimator m d is suitable to estimate the change over occasions. To device suitable estimation procedures for both the problems simultaneously, a convex linear combination of u d and m d has been taken as a final estimator of the population mean Y and is given by where   is a scalar to be determined such that the mean square error (MSE) of d is minimum. 4 ISSN 2424-6271 IASSL

Properties of the Proposed Estimator
To obtain the bias (B (.)) and mean square error (MSE (.)) of the estimator d, we write

The Bias and MSE of the Estimator
Taking expectation of both sides of (5) we get the bias of u d to the first degree of approximation as Remark -3.1 Since x and y denote the same study variable over two occasions, 1 z and 2 z are the stable auxiliary variables correlated the study variable x(y) , therefore , looking on the stability nature of the coefficients of variation [Murthy (1967, p.325)] and following Cochran (1977) and Feng and Zou (1977), the coefficients of variation of the variables x, y, 1 z and 2 z are considered to be approximately equal (i.e. .Thus under the Remark 3.1 the expression (6) reduces to: 6 ISSN 2424-6271 IASSL where ) ( Squaring both sides of (7) and neglecting terms of e's having power greater than two we have )] e e e e ( ) e e 2 e e )( 4 Taking expectation of both sides of (8) we get the MSE of u d to the first degree of approximation as Thus we have established the following theorem.

The Bias and MSE of the Estimator
Taking expectation of both sides of (14) we get the bias of m d to the first degree of approximation as Under the Remark 3.1 the bias of m d in (15) reduces to: Squaring both sides of (14) Taking expectation of both sides of (17) we get the MSE of m d to the first degree of approximation as where Now we state the following theorem.
For proof see Appendix-I.

Theorem 3.4 The mean squared error of ' d ' to the first degree of approximation is
given by

Optimum Replacement Policy
To obtain the optimum value of  (fraction of sample to be drawn a fresh on second occasion) so that the population mean Y may be estimated with maximum precision, we minimize the equation (30) with respect to  , which yields a quadratic equation in : (33)

Efficiency Comparisons
The percent relative efficiency of the proposed estimator ' d ' with respect to (i) n      . This behavior is in agreement with Sukhatme et al. (1984), results which explained that more the value of yx  , more the fractions of fresh sample required at the current occasion. (iii) Minimum value of 0  is 0.1994 (  0.20) , which shows that the fraction to be replaced at the current occasion is as low as about 20 percent of the total sample size leading to a reduction of considerable amount in the cost of the survey.
It is further observed from Table 1 to 4 that there is appreciable gain in efficiency by using the proposed estimator ' d ' over usual unbiased estimator n y and the estimator Ŷ . Thus we conclude that the use of auxiliary information at the estimation stage is highly rewarding in terms of the proposed estimator' d '.

Conclusion
In this article, we extend the current literature in two-occasion successive sampling using two auxiliary variables on both of the occasions and the information on study variable from the previous occasions one of which is positively correlated with the study variable while the other is negatively correlated. An efficient estimation procedure has been developed. Optimum replacement policy and the efficiency of the suggested estimator have been discussed. From the numerical study, it may be concluded that the proposed estimator is more beneficial in estimation of the population mean of the study variable at the current occasion in two occasion successive sampling. Finally, looking on the nice performance of the envisaged estimator, our recommendation is to use the proposed estimator by the survey practitioners in practice.