# Optimal rates for independence testing via $U$-statistic permutation tests

Berrett, TB and Kontoyiannis, I and Samworth, RJ Optimal rates for independence testing via $U$-statistic permutation tests. (Unpublished)

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## Abstract

We study the problem of independence testing given independent and identically distributed pairs taking values in a $\sigma$-finite, separable measure space. Defining a natural measure of dependence $D(f)$ as the squared $L_2$-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 $\{f: D(f) \geq \rho^2 \}$. 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 $D(f)$ 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.

Item Type: Article math.ST math.ST stat.ME stat.ML stat.TH 62C20, 62G10, 62H20 UNSPECIFIED Div F > Signal Processing and Communications Cron Job 22 Jan 2020 20:04 27 Oct 2020 06:10