Title: Differentiability of supremum-type functionals with applications
Authors: Javier Carcamo - Universidad Autonoma de Madrid (Spain) [presenting]
Antonio Cuevas - Autonomous University of Madrid (Spain)
Luis-Alberto Rodriguez - Universidad Autonoma de Madrid (Spain)
Abstract: The purpose is to 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 previous results 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. Finally, the asymptotic distribution of maximum mean discrepancies over Donsker classes of functions is derived.