Optimal rates for independence testing via U-statistic permutation tests

This is the authors' abstract. We don't add key points for this paper.

We study the problem of independence testing given independent and identically distributed pairs taking values in a σ-finite, separable measure space. Defining a natural measure of dependence D(f) as the squared L2-distance between a joint density f and the product of its marginals, we first show that there is no valid test of independence that is uniformly consistent against alternatives of the form {f:D(f)≥ρ2}. We therefore restrict attention to alternatives that impose additional Sobolev-type smoothness constraints, and define a permutation test based on a basis expansion and a U-statistic estimator of D(f) that we prove is minimax optimal in terms of its separation rates in many instances. Finally, for the case of a Fourier basis on [0,1]2, we provide an approximation to the power function that offers several additional insights. Our methodology is implemented in the R package USP.