We present a novel family of nonparametric omnibus tests of the hypothesis that two unknown but estimable functions are equal in distribution when applied to the observed data structure. We developed these tests, which represent a generalization of the maximum mean discrepancy tests described in Gretton et al. [2006], using recent developments from the higher-order pathwise differentiability literature. Despite their complex derivation, the associated test statistics can be expressed rather simply as U-statistics. We study the asymptotic behavior of the proposed tests under the null hypothesis and under both fixed and local alternatives. We provide examples to which our tests can be applied and show that they perform well in a simulation study. As an important special case, our proposed tests can be used to determine whether an unknown function, such as the conditional average treatment effect, is equal to zero almost surely.

中文翻译：

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对未知函数进行分布相等性的综合非参数测试。

我们提出了一个新的非参数综合检验族，该检验族的假设是，当将两个未知但可估计的函数应用于观察到的数据结构时，它们的分布相等。我们开发了这些测试，这些测试代表了Gretton等人中描述的最大平均差异测试的概括。[2006]，使用来自高阶路径可微性文献的最新进展。尽管它们的推导复杂，但是可以将相关的测试统计量表达为U统计量，而非常简单。我们研究在零假设下以及固定和局部替代条件下拟议测试的渐近行为。我们提供了可以应用我们的测试的示例，并显示了它们在模拟研究中的良好表现。作为一个重要的特殊情况，我们建议的测试可用于确定功能是否未知，