The maximum mean discrepancy (MMD) is a kernel-based nonparametric statistic for two-sample testing, whose inferential accuracy depends critically on variance characterization. Existing work provides various finite-sample estimators of the MMD variance, often differing under the null and alternative hypotheses and across balanced or imbalanced sampling schemes. In this paper, we study the variance of the MMD statistic through its U-statistic representation and Hoeffding decomposition, and establish a unified finite-sample characterization covering different hypotheses and sample configurations. Building on this analysis, we propose an exact acceleration method for the univariate case under the Laplacian kernel, which reduces the overall computational complexity from $\mathcal O(n^2)$ to $\mathcal O(n \log n)$.

翻译：最大均值差异（MMD）是一种基于核函数的非参数双样本检验统计量，其推断准确性关键取决于方差表征。现有研究提出了多种MMD方差的有限样本估计器，这些估计器通常在零假设与备择假设下存在差异，并在平衡或不平衡抽样方案中表现各异。本文通过MMD统计量的U统计量表示与Hoeffding分解研究其方差特性，建立了涵盖不同假设与样本配置的统一有限样本表征框架。基于此分析，我们针对拉普拉斯核下的单变量情形提出一种精确加速方法，将整体计算复杂度从$\mathcal O(n^2)$降低至$\mathcal O(n \log n)$。