Open Access
October, 1968 Approach to Degeneracy and the Efficiency of Some Multivariate Tests
G. K. Bhattacharyya, Richard A. Johnson
Ann. Math. Statist. 39(5): 1654-1660 (October, 1968). DOI: 10.1214/aoms/1177698147


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.


Download Citation

G. K. Bhattacharyya. Richard A. Johnson. "Approach to Degeneracy and the Efficiency of Some Multivariate Tests." Ann. Math. Statist. 39 (5) 1654 - 1660, October, 1968.


Published: October, 1968
First available in Project Euclid: 27 April 2007

zbMATH: 0197.15904
MathSciNet: MR230416
Digital Object Identifier: 10.1214/aoms/1177698147

Rights: Copyright © 1968 Institute of Mathematical Statistics

Vol.39 • No. 5 • October, 1968
Back to Top