On two methods of bias reduction in the estimation of ratios

Abstract

SUMMARY Quenouille's method of bias reduction, based on splitting the sample at random into groups, is applied to estimation of ratios and the optimum choice of the number of groups, g, is investigated assuming a model. It is shown that both the bias and variance of the estimator are decreasing functions of g. The estimator is compared with a modified ratio estimator assuming the same model. Durbin (1959) has applied Quenouille's (1956) method of bias reduction to the estimation of ratios. Suppose r denotes the customary ratio estimator of the form r = y/x based on n observations whose bias is en-' + O(n-2) where c is a constant and x and x are unbiased estimators of the population mean of the characters 'y' and 'x' respectively. Let the sample be divided at random into g groups, each of size p, where n = pg. Let rj denote the ratio estimator calculated from the sample after omitting the jth group (i.e. based on p(g -1) observations). Then the estimator (g-1) a 1 g rQ=ygr- 2 ~ rJ=- rQj, 1 Y j=i Y j=i