The two-sample problem for Poisson processes: Adaptive tests with a nonasymptotic wild bootstrap approach

Abstract

Considering two independent Poisson processes, we address the question of testing equality of their respective intensities. We first propose testing procedures whose test statistics are $U$-statistics based on single kernel functions. The corresponding critical values are constructed from a nonasymptotic wild bootstrap approach, leading to level $\alpha$ tests. Various choices for the kernel functions are possible, including projection, approximation or reproducing kernels. In this last case, we obtain a parametric rate of testing for a weak metric defined in the RKHS associated with the considered reproducing kernel. Then we introduce, in the other cases, aggregated or multiple kernel testing procedures, which allow us to import ideas coming from model selection, thresholding and/or approximation kernels adaptive estimation. These multiple kernel tests are proved to be of level $\alpha$, and to satisfy nonasymptotic oracle-type conditions for the classical $\mathbb{L} _{2}$-norm. From these conditions, we deduce that they are adaptive in the minimax sense over a large variety of classes of alternatives based on classical and weak Besov bodies in the univariate case, but also Sobolev and anisotropic Nikol’skii–Besov balls in the multivariate case.