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

Original · EN

Let (Xᵢ)ᵢ₌₁,...,ₙ be a possibly nonstationary sequence such that L(Xᵢ)=Pₙ if i≤ nθ and L(Xᵢ)=Qₙ 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 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.