We characterize the asymptotic performance of nonparametric goodness of fit testing. The exponential decay rate of the type-II error probability is used as the asymptotic performance metric, and a test is optimal if it achieves the maximum rate subject to a constant level constraint on the type-I error probability. We show that two classes of Maximum Mean Discrepancy (MMD) based tests attain this optimality on $mathbb R^d$, while the quadratic-time Kernel Stein Discrepancy (KSD) based tests achieve the maximum exponential decay rate under a relaxed level constraint. Under the same performance metric, we proceed to show that the quadratic-time MMD based two-sample tests are also optimal for general two-sample problems, provided that kernels are bounded continuous and characteristic. Key to our approach are Sanov’s theorem from large deviation theory and the weak metrizable properties of the MMD and KSD.
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机译：我们表征了拟合检验的非参数优良性的渐近性能。 II类错误概率的指数衰减率用作渐近性能度量，如果在I类错误概率受到恒定级别约束的情况下达到最大速率，则测试是最佳的。我们表明，基于最大平均差异（MMD）的两类测试在$ mathbb R ^ d $上达到了这种最优性，而基于二次时间内核斯坦因差异（KSD）的测试在宽松的水平约束下达到了最大指数衰减率。 。在相同的性能指标下，我们继续表明，基于二次方MMD的二次采样测试对于一般的二次采样问题也是最佳的，前提是内核具有连续性和特征性。我们方法的关键是来自大偏差理论的Sanov定理以及MMD和KSD的弱可度量性质。
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