An $${{\varvec{L}}}^{2}$$L2-norm-based test for equality of several covariance functions: a further study

For the multi-sample equal covariance function (ECF) testing problem, Zhang (Analysis of variance for functional data, CRC Press, Boca Raton, 2013) proposed an $$L^{2}$$L2-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 $$L^2$$L2-norm-based test is root-n consistent. In addition, intensive simulation studies are conducted to evaluate its finite sample performance against several existing competitors. In particular, they demonstrate that in terms of size control and power, the $$L^{2}$$L2-norm-based test outperforms the dimension-reduction-based tests proposed by Panaretos et al. (J Am Stat Assoc 105(490):670–682, 2010) and Fremdt et al. (Scand J Stat 40(1):138–152, 2013), respectively, when functional data are less correlated or when the covariance function differences of functional data contain mainly high-frequency signals. Two real data applications are presented to illustrate the $$L^2$$L2-norm-based test.