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
[Détection de rupture près des extrémités pour des suites fortements dépendantes]

Présenté par : 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.

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 :

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

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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},
     volume = {347},
     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/

Bibliographie
Cité par

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[2] S. Ben Hariz; J.J. Wylie; Q. Zhang Optimal rate of convergence for nonparametric change-point estimators for nonstationary sequences, Ann. Statist., Volume 35 (2007), pp. 1802-1826

[3] E. Carlstein Nonparametric change-point estimation, Ann. Statist., Volume 16 (1988), pp. 188-197

[4] M. Csörgő; L. Horváth Limit Theorems in Change-Point Analysis, Wiley, Chichester, 1997

[5] L. Dumbgen The asymptotic behavior of some nonparametric change-point estimators, Ann. Statist., Volume 19 (1991), pp. 1471-1495

[6] D. Ferger On the rate of almost sure convergence of Dumbgen's change-point estimators, Statist. Probab. Lett., Volume 19 (1995), pp. 27-31

[7] D. Ferger Exponential and polynomial tailbounds for change-point estimators, J. Statist. Plann. Inference, Volume 92 (2001), pp. 73-109

[8] D. Ferger Boundary estimation based on set-indexed empirical processes, Nonparametric Statist., Volume 16 (2004), pp. 245-260

[9] D.V. Hinkley Inference about the change-point in a sequence of random variables, Biometrika, Volume 57 (1970), pp. 1-17

[10] L. Horváth; P. Kokoszka The effect of long-range dependence on change-point estimators, J. Statist. Plann. Inference, Volume 64 (1997), pp. 57-81

[11] P. Kokoszka; R. Leipus Change-point in the mean of dependent observations, Statist. Probab. Lett., Volume 40 (1998), pp. 385-393

[12] W.L. Nie, S. Ben Hariz, J.J. Wylie, Q. Zhang, Detection of change-point near the end points of long-range dependent sequences, Preprint, 2008

[13] Y.-C. Yao; D. Huang; R.A. Davis On almost sure behavior of change-point estimators (E. Carlstein; H.-G. Müller; D. Siegmund, eds.), Change-Point Problems, IMS, Hayward, CA, 1994, pp. 359-372

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