Some asymptotic results for the integrated empirical process with applications to statistical tests

TL;DR

This paper investigates the strong approximation of the integrated empirical process, providing exact rates of approximation by weighted Brownian bridges and Kiefer processes, with applications to statistical testing and change-point detection.

Contribution

It offers new results on the strong approximation of the integrated empirical process, including when parameters are estimated, and explores applications to statistical tests and process behavior.

Findings

Derived exact approximation rates using weighted Brownian bridges.

Applied results to two-sample testing and change-point problems.

Analyzed the process behavior with estimated parameters.

Abstract

The main purpose of this paper is to investigate the strong approximation of the integrated empirical process. 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 the Koml\'os, Major and Tusn\'ady's results. Applications include the two-sample testing procedures together with the change-point problems. We also consider the strong approximation of the integrated empirical process when the parameters are estimated. Finally, we study the behavior of the self-intersection local time of the partial sum process representation of the integrated empirical process. Reference: Koml\'os, J., Major, P. and Tusn\'ady, G. (1975). An approximation of partial sums of independent RV's and the sample DF. I. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 32, 111-131.