Weighted Ratio-cum-Product Estimator for Finite Population Mean

Accepted 18/Aug/2018, Online 30/Aug/2018 Abstract We have, in this paper, proposed a new ratio-cum-product estimator of finite population mean using information on two auxiliary variates. The bias and mean squared error of the proposed estimator, up to the first order of approximation have been derived. The proposed estimator, under optimal weights, is shown to be superior to the competing estimators. Empirical investigations have been carried out in support of the theoretical findings.


INTRODUCTION
In survey sampling, a considerable attention is given for improving upon the usual unbiased estimator by the use of supplementary variable(s) in sampling theory and practice for estimation of population characteristics. The literature on survey sampling describes agreement of various techniques for utilizing information on auxiliary variate by ratio, product and regression methods of estimation to estimate the population parameters, out of which ratio and product are being easily obtainable and are more prevalent in practice.
Over the years, various estimators have been developed in simple random sampling to estimate the population characters using auxiliary information. Some noteworthy contributions in this area have been made by several authors including Hansen(1953), Olikin(1958), Goodman(1960), Koop(1964), Murthy(1964), Singh(1965), Singh(1966),  and many others.
In the present paper, we have studied some of the existing estimators for the population mean of a study variate by utilizing information on two auxiliary variates of which one is positively correlated with the study variate, while the other is negatively correlated. The proposed weighted ratio-cum-product estimator performs better than Singh's estimator, usual ratio and product estimators and simple unbiased estimatorunder practical conditions. We consider a finite population of size N, arbitrarily labelled 1, 2....N. Let Y and (X 1 , X 2 ) be the study and auxiliary variates, respectively, where X 1 is positively correlated with Y, while X 2 is negatively correlated with Y. Assuming that the population means and are known, a sample of size n (with n < N) is drawn from the population size N using simple random sampling without replacement (SRSWOR) scheme to estimate the population mean = ∑ of the study variate.
The whole paper is composed of six sections followed by references and authors profile. Section-I contains noteworthy contributions by several authors over the years to estimate population characters using auxiliary information. Section-II describes briefly some existing estimators by different authors. In section-III, the expressions of bias and mean square error of the proposed estimator have been derived, and also expression for optimum weight has been arrived at. Section-IV discusses efficiency comparisons of the proposed estimator with respect to competing estimators. Section-V deals with the empirical study to justify the supremacy of theoretical findings. Section-VI presents the summary and future directions.

II. REVIEW OF LITERATURE OF EXISTING ESTIMATORS
The variance of simple unbiased estimator is given by The usual ratio and product estimators of are, respectively, and ̅ ̅ ̅ ̅ whose biases and mean square errors, up to first degree of approximation, are, respectively, and where and are the coefficients of variations of and respectively, and .
M.P.  has suggested ratio-cum-product estimator, which is given by The bias and mean square error up to first degree of approximation are, respectively,

III. PROPOSED WEIGHTED RATIO-CUM-PRODUCT ESTIMATOR
We propose a weighted ratio-cum-product estimator for estimating the population mean , which is given by Let ⇒ = (1+ ), Putting the values of , and in the expression-(11), we get Upon simplification, we find that The bias of the proposed estimator, up to the first degree of approximation, is Similarly, the mean square error, to the first degree of approximation, is With a view to determining the most suitable value of , (and thus ), we proceed to minimize the mean square error subject to the variation in , implying thereby that

IV. EFFICIENCY COMPARISON
On comparison of (14) under optimum weights, with (10), we get Thus, the proposed weighted ratio-cum-product estimator fares better than the usual ratio-cum-product estimator if the above inequality holds good.

VI. CONCLUSION
The proposed weighted ratio-cum-product estimator has been shown to fare better than its competing estimators, e.g., , , , under practicalconditions. The theoretical developments have been numerically established. New ratio-cumproduct estimators can be designed using different weighting systems such as weighted geometric mean and weighted harmonic mean and their performance can be evaluated vis-a-vis the existing ratio-cumproduct estimators.