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. For the test thresholds, we derive a quantile bound for wild bootstrapped incomplete $U$- statistics, which is of independent interest. We derive 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 more widespread permutation-based approaches, 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 the three testing frameworks, we observe that our proposed linear-time aggregated tests obtain higher power than current state-of-the-art linear-time kernel tests.
翻译:我们提出了一系列计算效率高、非参数性测试,分别用于两个样本、独立和良好健康问题的计算标准。我们提出了一系列计算效率高、非参数性测试,分别使用最大偏差(MMDD)、Hilbert Schmich 独立标准(HSIC)和Kernel Stein Disception(KSD)等标准。我们的测试统计数据不全于美元统计数据,计算成本在样本数量的线性时间和与典型的美元统计测试相关的二次时间之间相互交叉。三个拟议测试在几个内核带带加在一起,以检测从不同尺度的空格差:我们称之为最终的MDAggInc、HSICAggInc和KSDAggInc。对于测试阈值的临界标准,我们得出了一种四分立的基点,在样本数量的线性时间之间,我们为MDAggInc和HSICAICGInc得出了统一的分离率,并精确地量化了我们计算效率与可实现的计算率之间的交易标准:这个结果是新结果,我们根据不全美的货币-美元货币货币货币货币货币货币货币的货币测试利率测试率测试方法,我们获得了我们各自的货币测试,在每件中,我们测测算的货币税率测试中,我们获得的货币税率测算。