# Some Asymptotic Results for the $p$-Fold Integrated Empirical Processes with Application to the Statistical Tests

Published in Test, 2018

### S. Alvarez-Andrade, S. Bouzebda and A. Lachal

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 the Komlós et al. (Z Wahrscheinlichkeitstheorie und Verw Gebiete 32:111–131, 1975)’s results. Applications include the two-sample testing procedures together with the change-point problems. Finally, simulation results are provided to illustrate the finite sample performance of the proposed statistical tests based on the integrated empirical processes.