Open Access
August 2007 Optimal rate of convergence for nonparametric change-point estimators for nonstationary sequences
Samir Ben Hariz, Jonathan J. Wylie, Qiang Zhang
Ann. Statist. 35(4): 1802-1826 (August 2007). DOI: 10.1214/009053606000001596


Let $( X_{i}) _{i=1,\ldots,n}$ be a possibly nonstationary sequence such that $\mathscr{L} (X_{i})=P_{n}$ if $i\leq n\theta $ and $\mathscr{L}(X_{i})=Q_{n}$ if $i \gt n\theta $, where $0 \lt \theta \lt 1$ 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 $\mathcal{F}$. 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, short-range dependent and as well long-range dependent sequences.


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Samir Ben Hariz. Jonathan J. Wylie. Qiang Zhang. "Optimal rate of convergence for nonparametric change-point estimators for nonstationary sequences." Ann. Statist. 35 (4) 1802 - 1826, August 2007.


Published: August 2007
First available in Project Euclid: 29 August 2007

zbMATH: 1147.62043
MathSciNet: MR2351106
Digital Object Identifier: 10.1214/009053606000001596

Primary: 60F99 , 62F05 , 62F10 , 62G20 , 62G30

Keywords: consistency , long-range dependence , nonparametric change-point estimation , nonstationary sequences , rates of convergence , short-range dependence

Rights: Copyright © 2007 Institute of Mathematical Statistics

Vol.35 • No. 4 • August 2007
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