Estimating a changed segment in a sample Alfredas Ra kauskas

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Estimating a changed segment in a sample? Alfredas Ra?kauskas Vilnius University and Institute of Mathematics and Informatics Department of Mathematics, Vilnius University Naugarduko 24, Lt-2006 Vilnius, Lithuania and Charles Suquet Laboratoire P. Painlevé (UMR 8524 CNRS) Bât. M2 U.F.R. de Mathématiques Université des Sciences et Technologies de Lille F-59655 Villeneuve d'Ascq Cedex France Abstract In the paper we consider a changed segment model for sample dis- tributions. We generalize Dümbgen's [6] change point estimator and obtain optimal rates of convergence of estimators of the begining and the length of the changed segment. Keywords : changed segment, changed segment location, empirical pro- cess, epidemic model, probability metrics, reproducing kernel. Mathematics Subject Classifications (2000): 62G05 1 Introduction A general changed segment (called also epidemic) model can be described as follows. For n = 3, 4, . . . , let Pn and Qn be two probability distributions on a measurable space E and let Xn,1, Xn,2, . . . , Xn,n be a triangular array of independent random elements in E. There exist s?n and t?n such that for 1 ≤ i ≤ ns?n or nt?n < i ≤ n, the Xi,n's have distribution Pn, while for ns?n < i ≤ nt?n, they have distribution Qn.

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Estimating

changed segment in a sample∗

AlfredasRačkauskas Vilnius University and Institute of Mathematics and Informatics Department of Mathematics, Vilnius University Naugarduko 24, Lt-2006 Vilnius, Lithuania and CharlesSuquet Laboratoire P. Painlevé (UMR 8524 CNRS) Bât. M2 U.F.R. de Mathématiques Université des Sciences et Technologies de Lille F-59655 Villeneuve d’Ascq Cedex France

Abstract

In the paper we consider a changed segment model for sample dis-tributions. We generalize Dümbgen’s [6] change point estimator and obtain optimal rates of convergence of estimators of the begining and the length of the changed segment.

Keywords :changed segment, changed segment location, empirical pro-cess, epidemic model, probability metrics, reproducing kernel.

Mathematics Subject Classiﬁcations (2000):62G05

Introduction

A general changed segment (called also epidemic) model can be described as follows. Forn= 3,4, . . ., letPnandQnbe two probability distributions on a measurable spaceEand letXn,1, Xn,2, . . . , Xn,nbe a triangular array of independent random elements inE exist. Theresn∗andt∗nsuch that for 1≤i≤nsn∗ornt∗n< i≤n, theXi,n’s have distributionPn, while for ns∗n< i≤nt∗n, they have distributionQn.We refer to e.g., Avery and Handerson [1], Commenges et al. [3], Račkauskas and Suquet [12], [13], ∗This work was supported by a cooperation agreement Lille-Vilnius EGIDE Gillibert.

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