# Directional differentiability for supremum-type functionals: statistical applications

## TL;DR

This paper establishes the directional differentiability of supremum-related functionals and applies these results to improve the understanding of the limiting distributions of various statistical tests, including Kolmogorov-Smirnov and copula-based statistics.

## Contribution

It introduces new Hadamard directional differentiability results for supremum-type functionals and extends their application to complex statistical tests and distributions.

## Key findings

- Derived the asymptotic distributions of supremum functionals.
- Extended results to multidimensional Kolmogorov-Smirnov tests.
- Solved an open problem on the Berk-Jones statistic.

## Abstract

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.