Zhu, Tianming

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 𝒳²-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 𝒳²-type mixtures, we employ the Welch–Satterthwaite 𝒳²-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 good performance of the proposed F-type test is also illustrated by a COVID-19 data example.

Multivariate functional data are prevalent in various fields such as biology, climatology, and finance. Motivated by the World Health Data applications, in this study, we propose and examine a global test for assessing the equality of multiple mean functions in multivariate functional data. This test addresses the one-way Functional Multivariate Analysis of Variance (FMANOVA) problem, which is a fundamental issue in the analysis of multivariate functional data. While numerous analysis of variance tests have been proposed and studied for univariate functional data, only a limited number of methods have been developed for the one-way FMANOVA problem. Furthermore, our global test has the ability to handle heteroscedasticity in the unknown covariance function matrices that underlie the multivariate functional data, which is not possible with existing methods. We establish the asymptotic null distribution of the test statistic as a chi-squared-type mixture, which depends on the eigenvalues of the covariance function matrices. To approximate the null distribution, we introduce a Welch–Satterthwaite type chi-squared-approximation with consistent parameter estimation. The proposed test exhibits root-𝓃 consistency, meaning it possesses nontrivial power against a local alternative. Additionally, it offers superior computational efficiency compared to several permutation-based tests. Through simulation studies and applications to the World Health Data, we highlight the advantages of our global test.