The Annals of Statistics

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

Samir Ben Hariz, Jonathan J. Wylie, and Qiang Zhang

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Abstract

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.

Article information

Source
Ann. Statist., Volume 35, Number 4 (2007), 1802-1826.

Dates
First available in Project Euclid: 29 August 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1188405631

Digital Object Identifier
doi:10.1214/009053606000001596

Mathematical Reviews number (MathSciNet)
MR2351106

Zentralblatt MATH identifier
1147.62043

Subjects
Primary: 60F99: None of the above, but in this section 62F10: Point estimation 62F05: Asymptotic properties of tests 62G20: Asymptotic properties 62G30: Order statistics; empirical distribution functions

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

Citation

Hariz, Samir Ben; Wylie, Jonathan J.; Zhang, Qiang. Optimal rate of convergence for nonparametric change-point estimators for nonstationary sequences. Ann. Statist. 35 (2007), no. 4, 1802--1826. doi:10.1214/009053606000001596. https://projecteuclid.org/euclid.aos/1188405631


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