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

Sergio Alvarez-Andrade (1, 2) , Salim Bouzebda (1, 2) , Aimé Lachal (3, 4)

1 LMAC - Laboratoire de Mathématiques Appliquées de Compiègne
2 UTC - Université de Technologie de Compiègne
3 PSPM - Probabilités, statistique, physique mathématique
4 INSA Lyon - Institut National des Sciences Appliquées de Lyon

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). 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. Finally, we study the behavior of the self-intersection local time of the partial sum process representation of integrated empirical processes.

Mots clés: Kiefer process; Brownian bridge; Integrated empirical process; Two-sample problem; Hypothesis testing; Local time; Rates of convergence; Goodness-of-fit; Change-point

Domaines: Statistiques [math.ST]; Probabilités [math.PR]

Sergio Alvarez-Andrade, Salim Bouzebda, Aimé Lachal. Strong approximations for the $p$-fold integrated empirical process with applications to statistical tests. Test, 2018, 27 (4), pp.826-849. ⟨10.1007/s11749-017-0572-0⟩. ⟨hal-01394849⟩