On a Further Robustness Property of the Test and Estimator Based on Wilcoxon's Signed Rank Statistic

Abstract

The robust-efficiency of the test and estimator based on Wilcoxon's [7] signed rank statistic when the sample observations are drawn from different populations is studied here. Let $X_1, \cdots, X_n$ be $n$ independent random variables distributed according to continuous cumulative distribution functions (cdf) $F_1(x), \cdots, F_n(x)$, respectively. Let $\mathscr{F}$ be the class of all continuous cdf's which are symmetric about their medians. If $F_1 = \cdots = F_n = F \varepsilon \mathscr{F}$, the Wilcoxon's [7] signed rank statistic provides a robust test for and estimator of the median of $F(x)$, (cf. [2], [4], [6], [7]). The asymptotic relative efficiency (ARE) of this test and estimator has been studied by Hodges and Lehmann [1]. The present investigation is concerned with the study of the robust-efficiency of the same when $F_1, \cdots, F_n$ are not necessarily identical.