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August 2020 Directional differentiability for supremum-type functionals: Statistical applications
Javier Cárcamo, Antonio Cuevas, Luis-Alberto Rodríguez
Bernoulli 26(3): 2143-2175 (August 2020). DOI: 10.3150/19-BEJ1188

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 Raghavachari (Ann. Statist. 1 (1973) 67–73) 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 Jager and Wellner (In A Festschrift for Herman Rubin (2004) 319–331 IMS). Finally, the asymptotic distribution of maximum mean discrepancies over Donsker classes of functions is derived.

Citation

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Javier Cárcamo. Antonio Cuevas. Luis-Alberto Rodríguez. "Directional differentiability for supremum-type functionals: Statistical applications." Bernoulli 26 (3) 2143 - 2175, August 2020. https://doi.org/10.3150/19-BEJ1188

Information

Received: 1 December 2018; Revised: 1 July 2019; Published: August 2020
First available in Project Euclid: 27 April 2020

zbMATH: 07193955
MathSciNet: MR4091104
Digital Object Identifier: 10.3150/19-BEJ1188

Rights: Copyright © 2020 Bernoulli Society for Mathematical Statistics and Probability

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Vol.26 • No. 3 • August 2020
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