Strong approximations for the $p$-fold integrated empirical process with applications to statistical tests

TL;DR

This paper establishes strong approximation results for the $p$-fold integrated empirical process using weighted Brownian bridges, with applications to statistical testing, change-point detection, and empirical process analysis.

Contribution

It provides the first exact rates of strong approximation for the $p$-fold integrated empirical process and explores applications to various statistical tests and processes.

Findings

Derived exponential bounds for tail probabilities.

Established strong approximation rates with weighted Brownian bridges.

Demonstrated finite sample performance through simulations.

Abstract

The main purpose of this paper is to investigate the strong approximation of the $p$-fold integrated empirical process, $p$ 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 $p$-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.