Tail behaviour for suprema of empirical processes
type: Item
instance of: scholarly article
title: Tail behaviour for suprema of empirical processes
zbMATH Open document ID: 0596.62053
author: Robert J. Adler
author: Lawrence Brown
published in: The Annals of Probability
publication date: 1986-01-01T00:00:00Z
review text: Let F be a continuous distribution function on the unit cube \(I^ k\) of \({\mathbb{R}}^ k\) with uniform univariate marginals, and let \(W_ F\) be the ''pinned Brownian sheet based on F'' (i.e. the centered Gaussian process with covariance \(EW_ F(x)W_ F(y)=F(x\wedge y)-F(x)F(y)\), \(x,y\in I^ k\). The normalized and centered empirical process based on F converges in law to \(W_ F\), and for this fact to be useful in testing hypotheses one needs estimates for the distribution of \(\sup_{t}W_ F(t).\) It is shown in this paper that for all \(\lambda\) and appropriate c, C, \[ c(F,k)\lambda^{2(k-1)}e^{-2\lambda^ 2}\leq P\{\sup_{t}W_ F(t)>\lambda \}\leq C(k)\lambda^{\quad 2(k-1)}c^{-2\lambda^ 2}, \] with the constants identified in the case \(k=2\). This type of results originated in \textit{J. Kiefer} and \textit{J. Wolfowitz}, [Trans. Am. Math. Soc. 87, 173-186 (1958; Zbl 0088.113)] and \textit{J. Kiefer} [Pac. J. Math. 11, 649-660 (1961; Zbl 0119.349)], and there is some recent literature (Goodman, Park and Skough, Cabaña and Wschebor) on the case of uniform F. The authors apply their results to obtain sharp upper and lower class results for suprema of empirical processes and mention that their unpublished manuscript (1985) contains applications to multivariate Kolmogorov-Smirnov tests.
Mathematics Subject Classification ID: 62G30
Mathematics Subject Classification ID: 60F10
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 62E20
Mathematics Subject Classification ID: 62E15
zbMATH DE Number: 3960785
zbMATH Keywords: tail behaviour
zbMATH Keywords: multivariate empirical processes
zbMATH Keywords: empirical
zbMATH Keywords: distribution function
zbMATH Keywords: Gaussian random fields
zbMATH Keywords: Brownian sheet
zbMATH Keywords: centered Gaussian process
zbMATH Keywords: upper and lower class results
zbMATH Keywords: multivariate Kolmogorov-Smirnov tests
Wikidata QID: Q105584332
reviewed by: Evarist Giné
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176992616
OpenAlex ID: W2067103866
DOI: 10.1214/AOP/1176992616
Recommended article: Tail behaviour for the suprema of Gaussian processes with applications to empirical processes
Recommended article: Tail bounds for the supremums of empirical processes over unbounded classes of functions
Recommended article: Tail asymptotic behavior of the supremum of a class of chi-square processes
Recommended article: Q5507211
Recommended article: Extreme value theory for suprema of random variables with regularly varying tail probabilities
Recommended article: An exponential inequality for suprema of empirical processes with heavy tails on the left
Recommended article: Tail asymptotics of supremum of certain Gaussian processes over threshold dependent random intervals
Recommended article: Exact tail asymptotics of the supremum attained by a Lévy process
Recommended article: Estimates of the supremum distribution for a certain class of random processes
Recommended article: A tail inequality for suprema of unbounded empirical processes with applications to Markov chains
description: scientific article; zbMATH DE number 3960785