Directional differentiability for supremum-type functionals: Statistical applications

L. A. Rodríguez Ramírez, J. Cárcamo Urtiaga, A. Cuevas González

Let X be a non-empty set and consider the space of bounded real functions on X equipped with the supremum norm. We denote the supremum of a bounded function q by s(q). Suppose that we wish to estimate s(q). If q can be estimated in the space of bounded functions, it is reasonable to use the plug-in estimator to approximate s(q). The aim of this work is dealing with the asymptotics of the plug-in estimator. To the best of our kowledge, the first result in this direction was obtained by Raghavachari in 1973. The proofs provided are essentially based on a careful analysis of the behaviour of the empirical process. However, we explore an alternative approach: the Functional Delta Method. In this work we analyze the Hadamard directional differentiability of s in order to apply the Functional Delta Method. As particular examples we show an extension of the results of Raghavachari and the answer of an open question about Berk-Jones statistic proposed by Jagger and Wellner in 2004.

Keywords: Berk-Jones statistic, empirical process, extended Functional Delta Method, Hadamard differentiability, Kolmogorov-Smirnov statistic

RM-1 Ramiro Melendreras Award
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