The general linear hypothesis testing problem for multivariate functional data with applications

TL;DR

This paper introduces a new global hypothesis test for multivariate functional data that effectively handles complex covariance structures and small sample sizes, with proven asymptotic properties and demonstrated practical performance.

Contribution

It develops a novel chi-squared-type mixture test for the GLHT problem in MFD, accommodating heteroscedasticity and correlation, and provides asymptotic and finite-sample validation.

Findings

Test accurately approximates the null distribution using a three-cumulant chi-squared method.

The test maintains effectiveness across various correlation structures and small samples.

Simulation and real data show broad applicability and strong performance.

Abstract

As technology continues to advance at a rapid pace, the prevalence of multivariate functional data (MFD) has expanded across diverse disciplines, spanning biology, climatology, finance, and numerous other fields of study. Although MFD are encountered in various fields, the development of methods for hypotheses on mean functions, especially the general linear hypothesis testing (GLHT) problem for such data has been limited. In this study, we propose and study a new global test for the GLHT problem for MFD, which includes the one-way FMANOVA, post hoc, and contrast analysis as special cases. The asymptotic null distribution of the test statistic is shown to be a chi-squared-type mixture dependent of eigenvalues of the heteroscedastic covariance functions. The distribution of the chi-squared-type mixture can be well approximated by a three-cumulant matched chi-squared-approximation with its approximation parameters estimated from the data. By incorporating an adjustment coefficient, the proposed test performs effectively irrespective of the correlation structure in the functional data, even when dealing with a relatively small sample size. Additionally, the proposed test is shown to be root-n consistent, that is, it has a nontrivial power against a local alternative. Simulation studies and a real data example demonstrate finite-sample performance and broad applicability of the proposed test.