  For the multi-sample equal covariance function (ECF) testing problem, Zhang
(2013) proposed an $L^{2}$-norm based test. However, its asymptotic power and
finite sample performance have not been studied. In this paper, its asymptotic
power is investigated under some mild conditions. It is shown that the
$L^2$-norm based test is root-$n$ consistent. In addition, intensive simulation
studies demonstrate that in terms of size-controlling and power, the
$L^{2}$-norm based test outperforms the dimension-reduction based test proposed
by Fremdt et al. (2013) when the functional data are less correlated or when
the effective signal information is located in high frequencies. Two real data
applications are also presented to demonstrate the good performance of the
$L^2$-norm based test.
