The uniform empirical process $U_n$ is considered as a process indexed by intervals. Powerful new exponential bounds are established for the process indexed by both "points" and intervals. These bounds trivialize the proof of the Chibisov-O'Reilly theorem concerning the convergence of the process with respect to $\|\cdot/q\|$-metrics and are used to prove an interval analogue of the Chibisov-O'Reilly theorem. A strong limit theorem related to the well-known Holder condition for Brownian bridge $U$ is also proved. Connections with related work of Csaki, Eicker, Jaeschke, and Stute are mentioned. As an application we introduce a new statistic for testing uniformity which is the natural interval analogue of the classical Anderson-Darling statistic.
Publié le : 1982-08-14
Classification:
$||cdot/q||$-metrics,
exponential bounds,
process convergence,
Anderson-Darling statistics,
Watson's statistic,
60F05,
60B10,
60G17,
62E20
@article{1176993773,
author = {Shorack, Galen R. and Wellner, Jon A.},
title = {Limit Theorems and Inequalities for the Uniform Empirical Process Indexed by Intervals},
journal = {Ann. Probab.},
volume = {10},
number = {4},
year = {1982},
pages = { 639-652},
language = {en},
url = {http://dml.mathdoc.fr/item/1176993773}
}
Shorack, Galen R.; Wellner, Jon A. Limit Theorems and Inequalities for the Uniform Empirical Process Indexed by Intervals. Ann. Probab., Tome 10 (1982) no. 4, pp. 639-652. http://gdmltest.u-ga.fr/item/1176993773/