Nonparametric multivariate rank tests and their unbiasedness

TL;DR

This paper develops new multivariate rank tests that are finite-sample unbiased and distribution-free, addressing a key gap in the statistical testing literature and comparing their power to existing methods.

Contribution

It introduces several novel rank tests for multivariate two-sample problems that are unbiased, distribution-free, and locally most powerful against specific alternatives.

Findings

Tests are finite-sample unbiased against broad alternatives

Tests are distribution-free under hypotheses and alternatives

Power comparisons show competitive performance with existing tests

Abstract

Although unbiasedness is a basic property of a good test, many tests on vector parameters or scalar parameters against two-sided alternatives are not finite-sample unbiased. This was already noticed by Sugiura [Ann. Inst. Statist. Math. 17 (1965) 261--263]; he found an alternative against which the Wilcoxon test is not unbiased. The problem is even more serious in multivariate models. When testing the hypothesis against an alternative which fits well with the experiment, it should be verified whether the power of the test under this alternative cannot be smaller than the significance level. Surprisingly, this serious problem is not frequently considered in the literature. The present paper considers the two-sample multivariate testing problem. We construct several rank tests which are finite-sample unbiased against a broad class of location/scale alternatives and are finite-sample distribution-free under the hypothesis and alternatives. Each of them is locally most powerful against a specific alternative of the Lehmann type. Their powers against some alternatives are numerically compared with each other and with other rank and classical tests. The question of affine invariance of two-sample multivariate tests is also discussed.