Detection of change-points near the end points of long-range dependent sequences

Nie, Weilin; Ben Hariz, Samir; Wylie, Jonathan; Zhang, Qiang

doi:10.1016/j.crma.2009.02.002

Statistics

Detection of change-points near the end points of long-range dependent sequences
[Détection de rupture près des extrémités pour des suites fortements dépendantes]

Présenté par : Philippe G. Ciarlet

Nie, Weilin ¹ ; Ben Hariz, Samir ² ; Wylie, Jonathan ³ ; Zhang, Qiang ³

¹ Department of Mathematics and statistics, Wuhan University, Wuhan, China
² Laboratoire de statistique et processus, département de mathématiques, Université du Maine, avenue Olivier-Messiaen, 72085 Le Mans cedex 9, France
³ Department of Mathematics, City University of Hong Kong, Kowloon Tong, Hong Kong

Comptes Rendus. Mathématique, Tome 347 (2009) no. 7-8, pp. 425-428.

Résumé
Abstract

On considère une suite d'observations ${(X_{i})}_{i = 1, \dots, n}$ avec des lois marginales vérifiant : $L (X_{i}) = P_{n}$ pour $i ⩽ n θ_{n}$ et $L (X_{i}) = Q_{n}$ pour $i > n θ_{n}$ . Le paramètre $θ_{n}$ , qui peut dépendre de la taille n de la suite d'observations, désigne la localisation du changement dans la loi marginale. On s'intéresse ici à l'estimation de ce paramètre. On considère le cas général où la position de rupture $θ_{n}$ peut converger vers l'une des deux extrémités de l'intervalle $[0, 1]$ lorsque la longueur de la suite tend vers l'infini. La suite peut être fortement dépendante, faiblement dépendante ou indépendante, voire même non stationnaire. On étudie une classe d'estimateurs non-paramètriques. On prouve qu'ils sont consistants et que leur vitesse de convergence est de $1 / n$ . On traite aussi le cas où la distance entre les distributions $P_{n}$ et $Q_{n}$ tend vers 0 quand n tend vers l'infini.

We consider a sequence of observations ${(X_{i})}_{i = 1, \dots, n}$ with a marginal distribution that is given by $L (X_{i}) = P_{n}$ if $i ⩽ n θ_{n}$ and $L (X_{i}) = Q_{n}$ if $i > n θ_{n}$ . The parameter $0 < θ_{n} < 1$ is the location of the change-point which must be estimated and may depend on the sequence length. We consider the general case in which the change-point can converge to one of the end-points of the interval $[0, 1]$ as the sequence length n tends to infinity. The sequence can be long-range dependent, short-range dependent or independent and may be non-stationary. We study a class of non-parametric estimators and prove they are consistent and that the rate of convergence is $1 / n$ . We also deal with the case in which the distance between the distributions $P_{n}$ and $Q_{n}$ tends to zero as n tends to infinity.

Reçu le : 2008-12-05
Accepté le : 2009-01-08
Publié le : 2009-03-17

DOI : 10.1016/j.crma.2009.02.002

Affiliations des auteurs :

Nie, Weilin ¹ ; Ben Hariz, Samir ² ; Wylie, Jonathan ³ ; Zhang, Qiang ³

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     author = {Nie, Weilin and Ben Hariz, Samir and Wylie, Jonathan and Zhang, Qiang},
     title = {Detection of change-points near the end points of long-range dependent sequences},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {425--428},
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Nie, Weilin; Ben Hariz, Samir; Wylie, Jonathan; Zhang, Qiang. Detection of change-points near the end points of long-range dependent sequences. Comptes Rendus. Mathématique, Tome 347 (2009) no. 7-8, pp. 425-428. doi : 10.1016/j.crma.2009.02.002. http://www.numdam.org/articles/10.1016/j.crma.2009.02.002/

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