  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.
