The two-sample problem for Poisson processes: adaptive tests with a non-asymptotic wild bootstrap approach

TL;DR

This paper develops adaptive non-asymptotic tests for comparing the intensities of two independent Poisson processes using wild bootstrap methods, achieving minimax optimality over various function classes.

Contribution

It introduces a novel adaptive testing procedure with non-asymptotic wild bootstrap calibration for the two-sample Poisson problem, covering a wide range of function spaces.

Findings

Tests are of level  and satisfy oracle inequalities.

The methods are adaptive over classical and weak Besov, Sobolev, and Nikol'skii-Besov spaces.

Achieves parametric rates under reproducing kernel Hilbert space assumptions.

Abstract

Considering two independent Poisson processes, we address the question of testing equality of their respective intensities. We first propose single tests whose test statistics are U-statistics based on general kernel functions. The corresponding critical values are constructed from a non-asymptotic 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, an aggregation procedure, which allows us to import ideas coming from model selection, thresholding and/or approximation kernels adaptive estimation. The resulting multiple tests are proved to be of level \alpha, and to satisfy non-asymptotic oracle type conditions for the classical L2-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.