Two-sample Behrens--Fisher problems for high-dimensional data: a normal reference F-type test

TL;DR

This paper introduces a new F-type test for high-dimensional two-sample mean vector equality problems that relaxes covariance matrix assumptions and demonstrates superior size control and practical applicability.

Contribution

The paper proposes a novel F-type test for high-dimensional Behrens--Fisher problems that does not require strong covariance assumptions and uses a normal reference distribution for testing.

Findings

The proposed F-type test outperforms existing methods in size control.

The test's null distribution can be approximated by an F-distribution with estimated degrees of freedom.

Simulation studies confirm the test's effectiveness and real data application demonstrates its utility.

Abstract

The problem of testing the equality of mean vectors for high-dimensional data has been intensively investigated in the literature. However, most of the existing tests impose strong assumptions on the underlying group covariance matrices which may not be satisfied or hardly be checked in practice. In this article, an F-type test for two-sample Behrens--Fisher problems for high-dimensional data is proposed and studied. When the two samples are normally distributed and when the null hypothesis is valid, the proposed F-type test statistic is shown to be an F-type mixture, a ratio of two independent chi-square-type mixtures. Under some regularity conditions and the null hypothesis, it is shown that the proposed F-type test statistic and the above F-type mixture have the same normal and non-normal limits. It is then justified to approximate the null distribution of the proposed F-type test statistic by that of the F-type mixture, resulting in the so-called normal reference F-type test. Since the F-type mixture is a ratio of two independent chi-square-type mixtures, we employ the Welch--Satterthwaite chi-square-approximation to the distributions of the numerator and the denominator of the F-type mixture respectively, resulting in an approximation F-distribution whose degrees of freedom can be consistently estimated from the data. The asymptotic power of the proposed F-type test is established. Two simulation studies are conducted and they show that in terms of size control, the proposed F-type test outperforms two existing competitors. The proposed F-type test is also illustrated by a real data example.