Kernels based tests with non-asymptotic bootstrap approaches for two-sample problems

Considering either two independent i.i.d. samples, or two independent samples generated from a heteroscedastic regression model, or two independent Poisson processes, we address the question of testing equality of their respective distributions. We first propose single testing procedures based on a general symmetric kernel. The corresponding critical values are chosen from a wild or permutation bootstrap approach, and the obtained tests are exactly (and not just asymptotically) of level. We then introduce an aggregation method, which enables to overcome the difficulty of choosing a kernel and/or the parameters of the kernel. We derive non-asymptotic properties for the aggregated tests, proving that they may be optimal in a classical statistical sense.