Some asymptotic results for the integrated empirical process with applications to statistical tests |
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| Authors: | Sergio Alvarez-Andrade Aimé Lachal |
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| Institution: | 1. Laboratoire de Mathématiques Appliquées de Compiègne, Sorbonne Universités, Université de Technologie de Compiègne, Cedex, France;2. Institut National des Sciences Appliquées de Lyon, Institut Camille Jordan, Université de Lyon, Cedex, France |
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| 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 Komlós, J., Major, P., Tusnády, G. (1975). An approximation of partial sums of independent RV's and the sample DF. I. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 32:111–131.Crossref], Web of Science ®] , Google Scholar])'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. |
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| Keywords: | Integrated empirical process Brownian bridge Kiefer process Rates of convergence Local time Two-sample problem Hypothesis testing Goodness-of-fit Change-point |
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