Kernel two-sample tests in high dimensions: interplay between moment discrepancy and dimension-and-sample orders
研究了当维度和样本量都趋于无穷时，核双样本检验的渐近行为，推导了中心极限定理，揭示了可检测的矩差异与维度和样本阶数之间的相互作用。
Summary Motivated by the increasing use of kernel-based metrics for high-dimensional and large-scale data, we study the asymptotic behaviour of kernel two-sample tests when the dimension and sample sizes both diverge to infinity. We focus on the maximum mean discrepancy using an isotropic kernel, which includes maximum mean discrepancy with the Gaussian kernel and the Laplace kernel, and the energy distance as special cases. We derive asymptotic expansions of the kernel two-sample statistics, based on which we establish a central limit theorem under both the null hypothesis and the local and fixed alternatives. The new nonnull central limit theorem results allow us to perform asymptotic exact power analysis, which reveals a delicate interplay between the moment discrepancy that can be detected by the kernel two-sample tests and the dimension-and-sample orders. The asymptotic theory is further corroborated through numerical studies.