Approach to Degeneracy and the Efficiency of Some Multivariate Tests

Abstract

When testing $p$-variate distributions for a shift in location, two important nonparametric competitors of Hotelling's $T{}^2$ are the multivariate extensions $W$ of the Wilcoxon test and $M$ of the normal score test. Bounds on their asymptotic relative efficiency (ARE) have been investigated by Hodges-Lehmann [6] and Chernoff-Savage [4] in the univariate case and by Bickel [3] and Bhattacharyya [1] in the multivariate case. The univariate normal score test has the commendable property that for all continuous distributions, its ARE with respect to the $t$-test exceeds 1 and with respect to the Wilcoxon test it exceeds $\pi/6$. This naturally raises the question of whether or not the multivariate extension $M$ inherits this property and if not, what the lower bounds on its ARE with respect to $W$ and $T{}^2$ are. In this paper, we answer this question by providing an example where the ARE of $M$ with respect to both $W$ and $T{}^2$ is arbitrarily close to zero for some direction. The example consists of a gross error distribution which places most of its mass on a hyperplane and has marginals with high sixth moments. Bickel [3] mentioned a similar property of the ARE of $W$ with respect to $T{}^2$. His proof for the case $p = 2$ is, however, incorrect. We show that for the type of gross error model considered by Bickel, the above ARE is bounded strictly away from zero. We correct his proof by constructing a distribution which also places high mass on a line but is not of the gross error type.