Mean of ratios or ratio of means or both?

Abstract

Consider a finite population of units ( U 1 , U 2 ,…, U N ). On each unit U i , variates of interest y and x are defined taking values Y i and X i , respectively, i =1,2,…, N . In certain surveys, it is of interest to estimate the population ratio R = Y / X (or equivalently, Y ̄ / X ̄ ), where Y =∑ 1 N Y i and X =∑ 1 N X i , based on a sample of size n selected according to a sampling design p ( s ). Under simple random sampling scheme, the usual choices for the estimation of R are (i) a (single) ratio of sample means given by R ̂ 1 = y ̄ / x ̄ or (ii) the mean of ( n ) ratios, viz. R ̂ n = ∑ 1 n (y i /x i )/n . It is well known that both R ̂ 1 and R ̂ n are biased for R . Using the extent of biases, we shall first discuss the role of R ̂ 1 and R ̂ n in the construction of unbiased ratio estimators. When y is considered as the study variate and x is an auxiliary variate related to y , the problem of estimation of the population mean Ȳ or the population total Y is dealt by constructing Y ̄ ̂ = R ̂ X ̄ or Y ̂ = R ̂ X . For the estimation of the population total Y , we shall consider a class of Symmetrized Des Raj (SDR) strategies and look for a choice of a model-optimum estimator when design-unbiasedness is not demanded, among those utilising ‘mean of ratios’ and ‘ratio of means’.