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

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 multivariate analysis of variance for functional data (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 asymptotic power of the proposed test under a local alternative is established. Simulation studies and a real data example demonstrate finite-sample performance and broad applicability of the proposed test.