论文标题
通过$ U $统计排列测试的独立测试的最佳率
Optimal rates for independence testing via $U$-statistic permutation tests
论文作者
论文摘要
我们研究了独立和相同分布对的独立性测试问题，以$σ$ - 罚款，可分开的度量空间进行值。将自然量度定义为$ d（f）$作为平方$ l^2 $ - 距离$ f $与边际产品的产物之间的距离，我们首先表明，没有有效的独立性测试，与形式$ \ \ {f：d（f）\ d（f）\ geq pect^2 \ \ fect off y sectionals均均无一致。因此，我们将注意力限制在对替代方案中施加额外的sobolev型平滑度约束，并根据基础扩展定义置换测试，在许多情况下，我们证明，我们证明我们证明这是最佳的$ d（f）$ $ d $ statistic估计器。最后，对于$ [0,1]^2 $的傅立叶基础，我们提供了功能功能的近似值，该功能提供了几种其他见解。我们的方法是在USP的R软件包中实现的。
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 $L^2$-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 $\{f: D(f) \geq ρ^2 \}$. 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.
