Statistics Papers

Document Type

Journal Article

Date of this Version

1986

Publication Source

The Annals of Probability

Volume

14

Issue

1

Start Page

1

Last Page

30

DOI

10.1214/aop/1176992616

Abstract

We consider multivariate empirical processes Xn(t):=√n(Fn(t)−F(t)), where Fn is an empirical distribution function based on i.i.d. variables with distribution function F and t ∈ Rk. For XF the weak limit of Xn, it is shown that

c(F,k)λ2(k−1)e−2λ^2P{suptXF(t) > λ} ≤ C(k) λ2(k−1)e−2λ^2

for large λ and appropriate constants c,C. When k = 2 these constants can be identified, thus permitting the development of Kolmogorov--Smirnov tests for bivariate problems. For general k the bound can be used to obtain sharp upper-lower class results for the growth of suptXn(t) with n.

Keywords

tail behaviour of suprema, empirical processes, Kolmogorov-Smirnov tests, Gaussian random fields

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Date Posted: 27 November 2017

This document has been peer reviewed.