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.

翻译：基于内核的测试提供了一个简单而有效的框架,它利用复制内核Hilbert空间的理论来设计非参数测试程序。在本文件中,我们提出了新的理论工具,可用于研究若干数据情景和许多不同的测试问题中内核测试的无症状行为。与目前的方法不同,我们的方法避免使用长期的美元和美元统计数据扩张和限制理论,这通常出现在文献中,并且直接与Hilbert空间的随机功能合作。因此,我们的框架导致对内核测试进行更简单、更清洁的分析,只需要温和的常规性条件。此外,我们表明,一般来说,通过证明我们的方法所要求的常规性条件既充分又必要,我们的分析是无法改进的。为了说明我们的方法的有效性,我们提出了对有条件独立测试问题进行新的内核测试,以及对已知的内核测试进行新的分析。