Optimal rates for independence testing via U-statistic permutation tests
Thomas B. Berrett, Ioannis Kontoyiannis, Richard J. Samworth
The Annals of Statistics · 2021 · Volume 49 · Issue 5
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 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.
Keywords
62G09, 62G10, Independence testing, minimax separation rates, permutation tests, Stein’s method, U-statistics
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