Strong approximations for the $p$-fold integrated empirical process with applications to statistical tests
Abstract
The main purpose of this paper is to investigate the strong approximation of the -fold integrated empirical process, being a fixed positive integer. More precisely, we obtain the exact rate of the approximations by a sequence of weighted Brownian bridges and a weighted Kiefer process. Our arguments are based in part on results of Koml\'os, Major and Tusn\'ady (1975). We also obtain an exponential bound for the tail probability of the weighted approximation to the -fold integrated empirical process. Applications include the two-sample testing procedures together with the change-point problems. We also consider the strong approximation of integrated empirical processes when the parameters are estimated. We study the behavior of the self-intersection local time of the partial sum process representation of integrated empirical processes. Finally, simulation results are provided to illustrate the finite sample performance of the proposed statistical tests based on the integrated empirical processes.
Cite
@article{arxiv.1611.04513,
title = {Strong approximations for the $p$-fold integrated empirical process with applications to statistical tests},
author = {Sergio Alvarez-Andrade and Salim Bouzebda and Aimé Lachal},
journal= {arXiv preprint arXiv:1611.04513},
year = {2019}
}
Comments
48 pages. This second version is a considerable extension of the first one. In particular, two new sections have been added (Sections 3 and 8) together with the corresponding proofs (Sections 9.9 and 9.10) and the corresponding references. An additional appendix (Appendix B) has been added too. arXiv admin note: substantial text overlap with arXiv:1505.07345