指数相容核双样本检验
Exponentially Consistent Kernel Two-Sample Tests
摘要：
给定来自未知分布$P$和$Q$的两组独立样本，双样本检验决定是否拒绝$P=Q$的原假设。最近的注意力集中在核两样本测试，因为检验统计量易于计算，收敛速度快，并且具有有限样本估计的低偏差。然而，仍然缺乏一个准确的表征渐近性能的测试，特别是，在大样本限制的第二类错误概率衰减到零的速度。在这项工作中，我们建立了一类核双样本检验是指数一致的波兰，局部紧Hausdorff样本空间，例如，$\mathbb R^d$。所得到的指数衰减率进一步被证明是最佳的所有两个样本测试满足水平约束，是独立的特定内核，只要它们是有界连续和特性。我们的研究结果获得了新的见解相关的问题，如公平的替代测试和内核选择策略。最后，作为一个应用程序，我们表明，基于内核的测试实现了最佳的检测离线变化检测的非参数设置。
英文摘要：
Given two sets of independent samples from unknown distributions $P$ and $Q$, a two-sample test decides whether to reject the null hypothesis that $P=Q$. Recent attention has focused on kernel two-sample tests as the test statistics are easy to compute, converge fast, and have low bias with their finite sample estimates. However, there still lacks an exact characterization on the asymptotic performance of such tests, and in particular, the rate at which the type-II error probability decays to zero in the large sample limit. In this work, we establish that a class of kernel two-sample tests are exponentially consistent with Polish, locally compact Hausdorff sample space, e.g., $\mathbb R^d$. The obtained exponential decay rate is further shown to be optimal among all two-sample tests satisfying the level constraint, and is independent of particular kernels provided that they are bounded continuous and characteristic. Our results gain new insights into related issues such as fair alternative for testing and kernel selection strategy. Finally, as an application, we show that a kernel based test achieves the optimal detection for off-line change detection in the nonparametric setting.
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