基于核的测试分析的一般框架
A general framework for the analysis of kernel-based tests
摘要：
基于核的测试提供了一个简单而有效的框架，使用再生核希尔伯特空间的理论来设计非参数测试程序。在本文中，我们提出了新的理论工具，可用于研究渐近行为的基于内核的测试在几个数据的情况下，在许多不同的测试问题。与目前的方法不同，我们的方法避免使用冗长的$U$和$V$统计扩展和极限定理，通常出现在文献中，并直接与随机泛函希尔伯特空间。因此，我们的框架导致了一个更简单和干净的内核测试分析，只需要温和的规则性条件。此外，我们表明，在一般情况下，我们的分析不能通过证明我们的方法所需的正则性条件是充分和必要的。为了说明我们方法的有效性，我们针对条件独立性测试问题提出了一个新的内核测试，以及对已知的基于内核的测试的新分析。
英文摘要：
Kernel-based tests provide a simple yet effective framework that use the theory of reproducing kernel Hilbert spaces to design non-parametric testing procedures. In this paper we propose new theoretical tools that can be used to study the asymptotic behaviour of kernel-based tests in several data scenarios, and in many different testing problems. Unlike current approaches, our methods avoid using lengthy $U$ and $V$ statistics expansions and limit theorems, that commonly appear in the literature, and works directly with random functionals on Hilbert spaces. Therefore, our framework leads to a much simpler and clean analysis of kernel tests, only requiring mild regularity conditions. Furthermore, we show that, in general, our analysis cannot be improved by proving that the regularity conditions required by our methods are both sufficient and necessary. To illustrate the effectiveness of our approach we present a new kernel-test for the conditional independence testing problem, as well as new analyses for already known kernel-based tests.
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