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As technology continues to advance at a rapid pace, the prevalence ofmultivariate 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 forhypotheses on mean functions, especially the general linear hypothesis testing(GLHT) problem for such data has been limited. In this study, we propose andstudy a new global test for the GLHT problem for MFD, which includes theone-way FMANOVA, post hoc, and contrast analysis as special cases. Theasymptotic null distribution of the test statistic is shown to be achi-squared-type mixture dependent of eigenvalues of the heteroscedasticcovariance functions. The distribution of the chi-squared-type mixture can bewell approximated by a three-cumulant matched chi-squared-approximation withits approximation parameters estimated from the data. By incorporating anadjustment coefficient, the proposed test performs effectively irrespective ofthe correlation structure in the functional data, even when dealing with arelatively small sample size. Additionally, the proposed test is shown to beroot-n consistent, that is, it has a nontrivial power against a localalternative. Simulation studies and a real data example demonstratefinite-sample performance and broad applicability of the proposed test.

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