  Testing the homogeneity between two samples of functional data is an
important task. While this is feasible for intensely measured functional data,
we explain why it is challenging for sparsely measured functional data and show
what can be done for such data. In particular, we show that testing the
marginal homogeneity based on point-wise distributions is feasible under some
constraints and propose a new two sample statistic that works well with both
intensively and sparsely measured functional data. The proposed test statistic
is formulated upon Energy distance, and the critical value is obtained via the
permutation test. The convergence rate of the test statistic to its population
version is derived along with the consistency of the associated permutation
test. To the best of our knowledge, this is the first paper that provides
guaranteed consistency for testing the homogeneity for sparse functional data.
The aptness of our method is demonstrated on both synthetic and real data sets.
