We propose a series of computationally efficient nonparametric tests for the two-sample, independence, and goodness-of-fit problems, using the Maximum Mean Discrepancy (MMD), Hilbert Schmidt Independence Criterion (HSIC), and Kernel Stein Discrepancy (KSD), respectively. Our test statistics are incomplete $U$-statistics, with a computational cost that interpolates between linear time in the number of samples, and quadratic time, as associated with classical $U$-statistic tests. The three proposed tests aggregate over several kernel bandwidths to detect departures from the null on various scales: we call the resulting tests MMDAggInc, HSICAggInc and KSDAggInc. This procedure provides a solution to the fundamental kernel selection problem as we can aggregate a large number of kernels with several bandwidths without incurring a significant loss of test power. For the test thresholds, we derive a quantile bound for wild bootstrapped incomplete $U$-statistics, which is of independent interest. We derive non-asymptotic uniform separation rates for MMDAggInc and HSICAggInc, and quantify exactly the trade-off between computational efficiency and the attainable rates: this result is novel for tests based on incomplete $U$-statistics, to our knowledge. We further show that in the quadratic-time case, the wild bootstrap incurs no penalty to test power over the more widespread permutation-based approach, since both attain the same minimax optimal rates (which in turn match the rates that use oracle quantiles). We support our claims with numerical experiments on the trade-off between computational efficiency and test power. In all three testing frameworks, the linear-time versions of our proposed tests perform at least as well as the current linear-time state-of-the-art tests.

翻译：我们针对两样本检验、独立性检验和拟合优度检验问题，分别利用最大均值差异（MMD）、希尔伯特-施密特独立准则（HSIC）和核斯坦因散度（KSD），提出一系列计算高效的非参数检验方法。所提出的检验统计量基于不完全$U$统计量，其计算代价在样本数线性时间和经典$U$统计量检验的二次时间之间插值。这三个检验通过聚合多个核带宽来检测不同尺度下对原假设的偏离：我们将相应的检验分别命名为MMDAggInc、HSICAggInc和KSDAggInc。该过程解决了核选择这一基本问题——我们能够聚合包含多个带宽的大量核，而不会导致检验功效显著损失。在检验阈值方面，我们推导了野生自助法不完全$U$统计量的分位数界，该结果具有独立研究价值。我们进一步推导了MMDAggInc和HSICAggInc的非渐近均匀分离率，并精确量化了计算效率与可达速率之间的权衡：据我们所知，这是针对基于不完全$U$统计量的检验的新成果。我们还证明，在二次时间情形下，野生自助法与更主流的基于置换的方法相比不会造成检验功效损失，因为两者可达到相同的极小极大最优速率（该速率与使用Oracle分位数时的速率一致）。我们通过数值实验验证了计算效率与检验功效之间的权衡关系。在所有三种检验框架中，我们所提出检验的线性时间版本至少与当前最优线性时间检验方法表现相当。