OPTIMAL RATES FOR INDEPENDENCE TESTING VIA U-STATISTIC PERMUTATION TESTS

• 成果类型: Article
• 署名作者: Berrett, Thomas B.; Kontoyiannis, Ioannis; Samworth, Richard J.
• 署名单位: University of Warwick; University of Cambridge
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/20-AOS2041
• 发表日期: 2021
• 页码: 2457-2490
• 关键词: CENTRAL-LIMIT-THEOREM; dependence; bootstrap

摘要

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) >= 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.