On the limiting distributions of multivariate depth-based rank sum statistics and related tests

TL;DR

This paper proves the conjectured limiting distribution of a multivariate depth-based rank sum statistic, validating its use in applications and demonstrating its potential for greater power than traditional tests in detecting distributional changes.

Contribution

It provides a proof of the conjectured limiting distribution for the multivariate depth-based rank sum statistic under general conditions.

Findings

The conjecture on the limiting distribution is proven.

The rank sum tests can outperform Hotelling's T^2 test.

The tests are effective in detecting location-scale changes.

Abstract

A depth-based rank sum statistic for multivariate data introduced by Liu and Singh [J. Amer. Statist. Assoc. 88 (1993) 252--260] as an extension of the Wilcoxon rank sum statistic for univariate data has been used in multivariate rank tests in quality control and in experimental studies. Those applications, however, are based on a conjectured limiting distribution, provided by Liu and Singh [J. Amer. Statist. Assoc. 88 (1993) 252--260]. The present paper proves the conjecture under general regularity conditions and, therefore, validates various applications of the rank sum statistic in the literature. The paper also shows that the corresponding rank sum tests can be more powerful than Hotelling's T^2 test and some commonly used multivariate rank tests in detecting location-scale changes in multivariate distributions.