Sharp-optimal adjustment for multiple testing in the multivariate two-sample problem

Abstract

Based on two independent samples $X_1, ...,X_m$ and $X_m+1, ...,X_n$ drawn from multivariate distributions with unknown Lebesgue densities p and q respectively, we propose an exact multiple test in order to identify simultaneously regions of significant deviations between p and q. The construction is built from randomized nearestneighbor statistics. It does not require any preliminary information about the multivariate densities such as compact support, strict positivity or smoothness and shape properties. The adjustment for multiple testing is sharp-optimal for typical arrangements of the observation values which appear with probability close to one, and it relies on a new coupling Bernstein type exponential inequality, reflecting the nonsubgaussian tail behavior of the combinatorial process. For power investigation of the proposed method a reparametrized minimax set-up is introduced, reducing the composite hypothesis ''$p = q$'' to a simple one with the multivariate mixed density $(m/n)p + (1 ? m/n)q$ as infinite dimensional nuisance parameter. Within this framework, the test is shown to be spatially and sharply asymptotically adaptive with respect to uniform loss on isotropic Hölder classes.