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

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

doi:10.1016/j.crma.2009.02.002

Statistics

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

Presented by: Philippe G. Ciarlet

Weilin Nie ¹; Samir Ben Hariz ²; Jonathan Wylie ³; Qiang Zhang 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, Volume 347 (2009) no. 7-8, pp. 425-428.

Abstract
Résumé

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.

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.

Received: 2008-12-05
Accepted: 2009-01-08
Published online: 2009-03-17

DOI: 10.1016/j.crma.2009.02.002

Author's affiliations:

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

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     author = {Weilin Nie and Samir Ben Hariz and Jonathan Wylie and Qiang Zhang},
     title = {Detection of change-points near the end points of long-range dependent sequences},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {425--428},
     publisher = {Elsevier},
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     number = {7-8},
     year = {2009},
     doi = {10.1016/j.crma.2009.02.002},
     language = {en},
}

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Weilin Nie; Samir Ben Hariz; Jonathan Wylie; Qiang Zhang. Detection of change-points near the end points of long-range dependent sequences. Comptes Rendus. Mathématique, Volume 347 (2009) no. 7-8, pp. 425-428. doi : 10.1016/j.crma.2009.02.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.02.002/

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