Optimal Rate of Convergence for Nonparametric Change-Point Estimators for Nonstationary Sequences

Let $(X_{i})_{i=1,\ldots ,n}$ be a possibly nonstationary sequence such that $\scr{L}(X_{i})=P_{n}$ if i . n. and $\scr{L}(X_{i})=Q_{n}$ if i > n., where 0 < . < 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 $\scr{F}$. 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.