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

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 Komlós et al. (1975)'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.

KEYWORDS:

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

MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 62G30
• 62G20
• 60F17
• Secondary: 62F03
• 62F12
• 60F15

Acknowledgments

The authors are grateful to the Editor, an Associate editor, and the referees for thorough proofreading, numerous comments, and several inspiring questions which led to a considerable improvement of the presentation.