Optimal rate of convergence for nonparametric change-point estimators for nonstationary sequences

Samir Ben Hariz; Jonathan J. Wylie; Qiang Zhang

doi:10.1214/009053606000001596

Author(s)

Related Research Unit(s)

Department of Mathematics

Detail(s)

Original language	English
Pages (from-to)	1802-1826
Journal / Publication	Annals of Statistics
Volume	35
Issue number	4
Publication status	Published - Aug 2007

Link(s)

DOI	DOI https://doi.org/10.1214/009053606000001596 Final Published version
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Attachment(s)	Documents 5554185.pdf Final Published version, 294 KB, PDF Publisher's Copyright Statement © Institute of Mathematical Statistics, 2007. Ben Hariz, S., Wylie, J. J., & Zhang, Q. (2007). Optimal rate of convergence for nonparametric change-point estimators for nonstationary sequences. Annals of Statistics, 35(4), 1802-1826. https://doi.org/10.1214/009053606000001596
Link to Scopus	https://www.scopus.com/record/display.uri?eid=2-s2.0-41849138696&origin=recordpage
Permanent Link	https://scholars.cityu.edu.hk/en/publications/publication(cd61c036-8a3e-48ac-8f90-03959cfe8a68).html

Abstract

Let (X_i)_i=1,...,n be a possibly nonstationary sequence such that script L(X_i) = P_n if i ≤ nθ and script L(X_i) = Q_n if i > nθ, where 0 <θ <1 is the location of the change-point to be estimated. We construct a class of estimators based on the empirical measures and a seminorm on the space of measures defined through a family of functions script F. We prove the consistency of the estimator and give rates of convergence under very general conditions. In particular, the 1/n rate is achieved for a wide class of processes including long-range dependent sequences and even nonstationary ones. The approach unifies, generalizes and improves on the existing results for both parametric and nonparametric change-point estimation, applied to independent, shortrange dependent and as well long-range dependent sequences. © Institute of Mathematical Statistics, 2007.