A Two Sample Test based on U-statistic for Functional Data

We propose a two-sample test for the mean functions of functional data when the number of bases is much lager than the sample size. The novel test is based on U-statistics which avoids estimating the covariance operator accurately under the high dimensional situation. We further prove the asymptotic normality of our test statistic under both null hypothesis and a local alternative hypothesis. An extensive simulation study is presented which shows that the proposed test works well in comparison with several other methods under the high dimensional situation. An application to egg-laying trajectories of Mediterranean fruit flies data set demonstrates the applicability of the method.