Re-Inforced Mann-Whitney Tests for the Two-Sample Location Problem

Abstract

Let S_x : X₁, X₂,......X _{n 1}, and S_y : Y₁, Y₂,......Y _{n 2} be independent random samples from two populations with densities f(x) and f(x-θ), where f(.) is unspecified having a finite Fisher information and θ is an unknown location parameter. We wish to test H_o : θ=0. In this paper a test has been proposed by making use of the information contained hi the sample order statistics in addition to that in the ranks of the observations. The innovation consists in trichotomizing each sample at two sample quanntiles of order p₁ , p₂, (0 <p₁ <p₂) and then forming a Mann-Whitney like statistic corresponding to each combination of one sub-sample from the first and one from the second. The test criterion is an optimum linear compound of these component statistics. The compounding coefficients ofcourse are functionals of the null density f(x) and as such have to be estimated from the sample. We have proposed consistent estimates of these and the resulting adaptive test has been proved to be asymptotically at least as powerful as the Mann- Whitney test. The gain in efficiency is found to be appreciable when the parent density has tail-weight either mucl;llighter or much heavier than that of the logistic density.