Testing the equality of several covariance functions for functional data: A supremum-norm based test

Testing the equality of covariance functions is crucial for solving functional ANOVA problems. Available methods, such as the recently proposed L2-norm based tests work well when functional data are less correlated but are less powerful when functional data are highly correlated or with some local spikes, which are often the cases in real functional data analysis. To overcome this difficulty, a new test for the equality of several covariance functions is proposed. Its test statistic is taken as the supremum value of the sum of the squared differences between the estimated individual covariance functions and the pooled sample covariance function. The asymptotic random expressions of the test statistic under the null hypothesis and under a local alternative are derived and a non-parametric bootstrap method is suggested. The root-n consistency of the proposed test is also obtained. Intensive simulation studies are conducted to demonstrate the finite sample performance of the proposed test. The simulation results show that the proposed test is indeed more powerful than several existing L2-norm based competitors when functional data are highly correlated or with some local spikes. The proposed test is illustrated with three real data examples collected in a wide scope of scientific fields.