论文标题
使用不完整的$ u $统计数据有效的聚合核测试
Efficient Aggregated Kernel Tests using Incomplete $U$-statistics
论文作者
论文摘要
我们建议使用最大平均差异（MMD），Hilbert Schmidt独立标准（HSIC）和内核Stein差异（KSD）提出一系列针对两样本，独立性和拟合优点的计算有效的非参数测试。我们的测试统计数据是不完整的$ u $统计信息，其计算成本与与经典$ u $ u $统计测试相关的样本数量和二次时间之间的线性时间之间进行了插值。这三个提出的测试在几个内核带宽上汇总，以检测各种尺度上的零：我们称之为结果测试mmdagginc，hsicagginc和ksdagginc。此过程为基本内核选择问题提供了解决方案，因为我们可以在不产生大量测试能力的情况下汇总许多带有多个带宽的内核。对于测试阈值，我们得出了一个巨大的自举不完整的$ U $统计量的分位数，这是一个独立的关注。我们得出了MMDagginc和Hsicagginc的非反应均匀分离率，并准确量化了计算效率和可达到的速率之间的权衡：据我们所知，基于不完整的$ U $阶段的测试是新颖的。我们进一步表明，在二次时间案例中，野生引导程序对基于更广泛的基于置换的方法进行测试电源没有损失，因为两者都达到了相同的最小最佳速率（这反过来又与使用Oracle分位数的速率相匹配）。我们通过数值实验对计算效率和测试能力之间的权衡进行数字实验来支持我们的主张。在所有三个测试框架中，我们提出的测试的线性时间版本至少以及当前的线性时间最新测试。
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.
