On the Optimality of Kernel-Embedding Based Goodness-of-Fit Tests
On the Optimality of Kernel-Embedding Based Goodness-of-Fit Tests
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发表时间：
2017-09
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通讯作者：
K. Balasubramanian;Tong Li;M. Yuan
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作者：
K. Balasubramanian;Tong Li;M. Yuan
分布的再现核希尔伯特空间（RKHS）嵌入为测试任意域中的问题提供了通用且灵活的框架，并且近年来引起了相当多的关注。为了深入了解它们的操作特性，我们在这里研究了此类方法在极小极大框架内的统计性能。着眼于拟合优度测试的情况，我们的分析表明，基于内核嵌入的测试的普通版本可能不是最佳的，并建议通过调节嵌入来进行简单的补救措施。我们证明，调节方法为与零值的大范围偏差提供了最佳测试，并且还可以在大量插值空间上进行自适应。数值实验进一步证明了我们方法的优点。
The reproducing kernel Hilbert space (RKHS) embedding of distributions offers a general and flexible framework for testing problems in arbitrary domains and has attracted considerable amount of attention in recent years. To gain insights into their operating characteristics, we study here the statistical performance of such approaches within a minimax framework. Focusing on the case of goodness-of-fit tests, our analyses show that a vanilla version of the kernel-embedding based test could be suboptimal, and suggest a simple remedy by moderating the embedding. We prove that the moderated approach provides optimal tests for a wide range of deviations from the null and can also be made adaptive over a large collection of interpolation spaces. Numerical experiments are presented to further demonstrate the merits of our approach.