We consider multivariate empirical processes $X_n(t) := \sqrt n (F_n(t) - F(t))$, where $F_n$ is an empirical distribution function based on i.i.d. variables with distribution function $F$ and $t \in \mathbb{R}^k$. For $X_F$ the weak limit of $X_n$, it is shown that $c(F, k)\lambda^{2(k-1)}e^{-2\lambda^2} \leq P\big\{\sup_t X_F(t) > \lambda\big\} \leq C(k)\lambda^{2(k-1)}e^{-2\lambda^2}$ for large $\lambda$ 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 $\sup_tX_n(t)$ with $n$.
Publié le : 1986-01-14
Classification:
Tail behaviour of suprema,
empirical processes,
Kolmogorov-Smirnov tests,
Gaussian random fields,
62G30,
60F10,
60F15,
62E20
@article{1176992616,
author = {Adler, Robert J. and Brown, Lawrence D.},
title = {Tail Behaviour for Suprema of Empirical Processes},
journal = {Ann. Probab.},
volume = {14},
number = {4},
year = {1986},
pages = { 1-30},
language = {en},
url = {http://dml.mathdoc.fr/item/1176992616}
}
Adler, Robert J.; Brown, Lawrence D. Tail Behaviour for Suprema of Empirical Processes. Ann. Probab., Tome 14 (1986) no. 4, pp. 1-30. http://gdmltest.u-ga.fr/item/1176992616/