上确界型泛函的方向可微性：统计应用
Directional differentiability for supremum-type functionals: statistical applications
摘要：
我们证明了定义在任意集合或测度空间上的实函数的上确界所涉及的各种泛函都是Hadamard方向可微的。我们特别考虑了上确界范数、上确界、下确界和函数的振幅。这些映射的（通常是非线性的）导数在适当的假设下采用简单的表达式。作为应用，我们改进并推广了\cite{Raghavachari}中关于在备择假设下Kolmogorov-Smirnov型统计量极限分布的结果.类似的结果得到类似的统计与copula。我们还解决了一个公开的问题有关的伯克-琼斯统计提出的\cite{Jager-Wellner-2004}。最后，给出了Donsker函数类上最大平均差的渐近分布。
英文摘要：
We show that various functionals related to the supremum of a real function defined on an arbitrary set or a measure space are Hadamard directionally differentiable. We specifically consider the supremum norm, the supremum, the infimum, and the amplitude of a function. The (usually non-linear) derivatives of these maps adopt simple expressions under suitable assumptions on the underlying space. As an application, we improve and extend to the multidimensional case the results in \cite{Raghavachari} regarding the limiting distributions of Kolmogorov-Smirnov type statistics under the alternative hypothesis. Similar results are obtained for analogous statistics associated with copulas. We additionally solve an open problem about the Berk-Jones statistic proposed by \cite{Jager-Wellner-2004}. Finally, the asymptotic distribution of maximum mean discrepancies over Donsker classes of functions is derived.
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