  In this paper, we propose two new tests for testing the equality of the
covariance functions of several functional populations, namely a quasi GPF test
and a quasi $F_{\max}$ test. The asymptotic random expressions of the two tests
under the null hypothesis are derived. We show that the asymptotic null
distribution of the quasi GPF test is a chi-squared-type mixture whose
distribution can be well approximated by a simple scaled chi-squared
distribution. We also adopt a random permutation method for approximating the
null distributions of the quasi GPF and $F_{\max}$ tests. The random
permutation method is applicable for both large and finite sample sizes. The
asymptotic distributions of the two tests under a local alternative are
investigated and they are shown to be root-n consistent. Simulation studies are
presented to demonstrate the finite-sample performance of the new tests against
three existing tests. They show that our new tests are more powerful than the
three existing tests when the covariance functions at different time points
have different scales. An illustrative example is also presented.
