Optimal rates for independence testing via U-statistic permutation tests

Abstract

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 $\mathit{D}(\mathit{f})$ as the squared ${\mathit{L}}^2$-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 $\{\mathit{f}:\mathit{D}(\mathit{f})\ge {\mathit{\rho }}^2\}$. 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 $\mathit{D}(\mathit{f})$ 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 $\mathtt{R}$ package $\mathtt{USP}$.