On Sampling Schemes Providing Unbiased Ratio Estimators

Abstract

The problem of finding unbiased ratio estimators of the population total of some character with the help of an auxiliary character has drawn much attention in recent years. Some references to this are given at the end. Under commonly adopted sampling schemes Hartley and Ross (1954), Goodman and Hartley (1958), Mickey (1959) and others derived certain unbiased ratio type estimators of the population total, while Lahiri (1951), Midzuno (1952), Des Raj (1954), Nanjamma, Murthy, and Sethi (1959) and others gave modifications of certain sampling schemes under which the ratio estimators of current type were unbiased. The first group of authors was primarily concerned with getting new types of unbiased ratio estimators under common sampling schemes. However, the second group was concerned with introducing little modifications in certain sampling schemes so that the usual ratio estimators of these sampling schemes become unbiased under the modified sampling schemes. Certain extensions in the latter direction are given in this paper. No comparison concerning the relative merits and demerits of the two methods of getting unbiased ratio type estimators is attempted here. Nanjamma, Murthy, and Sethi (1959) have given a general procedure of unbiased estimation of certain type of parameters such as population total and variance etc., and have shown how a given sampling scheme can be modified to make ratio estimators of such parameters unbiased. The modification can be applied to most of the sampling schemes commonly met in practice. In this paper it is shown that if in these modified sampling schemes a sufficient statistic is available [as is usually the case in with replacement sampling schemes, reference Basu (1958) and Pathak (1962(a), (b))] and if the ratio estimator does not depend on the sufficient statistic, it can be uniformly improved by Rao-Blackwell theorem. This result has been used to derive ratio estimators better than the ratio estimators given by Nanjamma, Murthy and Sethi. It is shown that in these modified sampling schemes the improved ratio estimator is the ratio of improved estimators of the numerator to the denominator derived under the original sampling scheme (without modification). Application of this result is given to some commonly used sampling schemes.