Global tests for detecting change in mean vector functions of multivariate functional data with repeated observations

In many scientific and technological fields, multivariate functional data are often repeatedly observed under varying conditions over time. A fundamental question is whether the mean vector function remains consistently equal throughout the entire period. This paper introduces two novel global testing statistics that leverage integration technique to address this issue. The asymptotic distributions of the proposed test statistics under the null hypothesis are derived, and their root-$n$ consistency is established. Simulation studies are conducted to evaluate the numerical performance of the proposed tests, which are further illustrated through an analysis of publicly available EEG motion data.