Tests for equality of several covariance matrix functions for multivariate functional data

Multivariate functional data are often observed in many scientific fields. This paper considers a multi-sample equal-covariance matrix function testing problem for multivariate functional data. Two new tests are proposed and studied. The asymptotic properties of the two tests under the null hypothesis and a local alternative are investigated. Two methods for approximating the null distributions of the test statistics are described. It is shown that the two tests are root-$n$ consistent. Two simulation studies are conducted to evaluate the finite sample performance of the proposed tests. Finally, the two tests are illustrated via applications to three real multivariate functional data sets.