We propose a family of maximum mean discrepancy (MMD) kernel two-sample tests that have low sample complexity and are consistent. The test has a hyperparameter that allows one to control the tradeoff between sample complexity and computational time. Our family of tests, which we denote as B-tests, is both computationally and statistically efficient, combining favorable properties of previously proposed MMD two-sample tests. It does so by better leveraging samples to produce low variance estimates in the finite sample case, while avoiding a quadratic number of kernel evaluations and complex null-hypothesis approximation as would be required by tests relying on one sample U-statistics. The B-test uses a smaller than quadratic number of kernel evaluations and avoids completely the computational burden of complex null-hypothesis approximation, while maintaining consistency and probabilistically conservative thresholds on Type I error. Finally, recent results of combining multiple kernels transfer seamlessly to our hypothesis test, allowing a further increase in discriminative power and decrease in sample complexity.

引入了一族最大平均差异(MMD)内核双样本检验，其中家族成员被称为 Block-tests 或 B-tests，检验统计量是样本子集上计算的 MMD 的平均，块大小的选择允许在测试功率和计算时间之间的权衡。B-test 家族结合了先前建议的 MMD 两个样本检验的有利特性：B-tests比样本对成对大小的线性时间测试更强大，但它们比使用单个大块计算U统计量的二次时间测试更具计算效率。B-tests的另一个重要优势是它们的渐近正常空分布，这与U统计量形成对比，在零假设下是退化的，计算零分布的估计是具有计算要求的。最近的关于假设检验的内核选择的结果可以无缝转移到B-tests，通过内核选择优化测试功率的方法。

B-tests: 低方差核双样本检验