Testing distributional equality for functional random variables

In this article, we propose a two-sample test for functional observations modeled as elements of a separable Hilbert space. We present a general recipe for constructing a measure of dissimilarity between the distributions of two Hilbertian random variables and study the theoretical properties of one such measure which is constructed using Maximum Mean Discrepancy (MMD) on random linear projections of the distributions and aggregating them. We propose a data-driven estimate of this measure and use it as the test statistic. Large sample distributions of this statistic are derived both under null and alternative hypotheses. This test statistic involves a kernel function and the associated bandwidth. We prove that the resulting test has large-sample consistency for any data-driven choice of bandwidth that converges in probability to a positive number. Since the theoretical quantiles of the limiting null distribution are intractable, in practice, the test is calibrated using the permutation method. We also derive the limiting distribution of the permuted test statistic and the asymptotic power of the permutation test under local contiguous alternatives. This shows that the permutation test is consistent and statistically efficient in the Pitman sense. Extensive simulation studies are carried out and a real data set is analyzed to compare the performance of our proposed test with some state-of-the-art methods.