On the Trimmed Mann-Whitney Statistic

Abstract

Consider random samples from two independent distributions with absolutely continuous distribution functions $F(z)$ and $F(z - \theta)$, respectively. For testing the hypotheses $\theta = 0$ against $\theta > 0$, Hodges and Lehmann [6] show that the Pitman asymptotic efficiency of $W$, the Mann-Whitney form of the Wilcoxon statistic, with respect to the $t$-statistic is never smaller than .864 and in their 4th Berkeley Symposium paper [7] they indicate this efficiency is almost always greater than or equal to 1 for distributions with tails at least as heavy as those of a normal distribution. Hence for distributions with heavier tails, $W$ is a more robust statistic than the $t$-statistic. For the moment, consider a single sample of size $n$ from a distribution with absolutely continuous distribution function $F(z - \theta)$. For distributions with heavier tails some authors have proposed the $\alpha$-trimmed mean as an estimate of $\theta$; that is, the mean based on the middle $n - 2\lbrack n\alpha\rbrack$ observations. Tukey [16] and Huber [8] study this statistic when $F$ is a contaminated normal distribution. Bickel [3] studies the asymptotic relative efficiency properties of the $\alpha$-trimmed mean relative to the mean for the class of continuous distributions with symmetric, unimodal densities. For estimating the location of a Cauchy distribution, Rothenberg, Fisher and Tilanus [12] show the trimmed mean based on the middle 24 percent of the observations is the most efficient trimmed mean relative to the maximum likelihood estimate. This single sample statistic suggests a Mann-Whitney statistic based on trimmed samples. It is hoped that the effects of contamination by gross errors in the underlying distributions can be considerably reduced by considering such a statistic. We have the benefits of using a simple and well known rank statistic and, at the same time, of being able to increase the efficiency by adjusting the trimming proportions according to the weight in the tails of the underlying distributions. This type of statistic is also related to rank tests for censored data which have been studied by Basu [2], Gastwirth [5] and Sobel [14], [15]. If we have samples size $m$ and $n$ from absolutely continuous distributions corresponding to $F(z)$ and $F(z - \theta)$, respectively, we denote by $W_\alpha$ the Mann-Whitney statistic based on the middle $m - 2\lbrack m\alpha\rbrack$ and $n - 2\lbrack n\alpha\rbrack$ observations of the samples. We refer to $W_\alpha$ as the $\alpha$-trimmed Mann-Whitney statistic. It is the purpose of this paper to investigate the Pitman asymptotic relative efficiency properties of $W_\alpha$ for a sub-class of absolutely continuous distributions. First some definitions are given in Section 2. In Section 3 and Section 4 we establish the asymptotic normality of $W_\alpha$, derive the efficiency of $W_\alpha$ relative to $W$ and give some examples. The results of Section 5 include a greatest lower bound on this efficiency. It is interesting to note that this bound is the same one found by Bickel [3] for the efficiency of the trimmed mean relative to the mean.