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

Let (X i ) ι =1 be a possibly nonstationary sequence such that L(X i ) = P n if i ≤ n6 and L(X i ) = Q n if > 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 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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