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

摘要

Let (X-i)(i)=1,..., n be a possibly nonstationary sequence such that L(X-i) = P-n, if i <= n theta and 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 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 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.