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

Optimal rate of convergence for nonparametric change-point estimators for nonstationary sequences. (English) Zbl 1147.62043

Ann. Stat. 35, No. 4, 1802-1826 (2007).
Summary: Let \((X_i)_{i=1,\dots,n}\) be a possibly nonstationary sequence such that \({\mathcal L}(X_i)= P_n\) if \(i\leq n\theta\) and \({\mathcal L}(X_i)= Q_n\) if \(i>n\theta\), where \(0<\theta<1\) is the location of the change-point to be estimated. We construct a class of estimators based on 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 existing results for both parametric and nonparametric change-point estimation, applied to independent, short-range dependent and as well long-range dependent sequences.

Cited in 19 Documents

MSC:

62G20 Asymptotic properties of nonparametric inference
62G05 Nonparametric estimation
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62F12 Asymptotic properties of parametric estimators
62G30 Order statistics; empirical distribution functions

Keywords:

long-range dependence; short-range dependence; consistency
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References:

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