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

• URL: http://arxiv.org/abs/2001.05513v2
• Date: Fri, 6 Nov 2020 11:50:28 GMT
• Title: Optimal rates for independence testing via $U$-statistic permutation tests
• Authors: Thomas B. Berrett, Ioannis Kontoyiannis, Richard J. Samworth
• Abstract summary: We study the problem of independence testing given independent and identically distributed pairs taking values in a $sigma$-finite, separable measure space. We first show that there is no valid test of independence that is uniformly consistent against alternatives of the form $f: D(f) geq rho2 $.
• Score: 7.090165638014331
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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. Our methodology is implemented in the R package USP.

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