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

Abstract

Let $( X_{i}) _{i=1,\ldots,n}$ be a possibly nonstationary sequence such that $\mathscr{L} (X_{i})=P_{n}$ if $i\leq n\theta $ and $\mathscr{L}(X_{i})=Q_{n}$ if $i \gt n\theta $, where $0 \lt \theta \lt 1$ 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 $\mathcal{F}$. 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 the existing results for both parametric and nonparametric change-point estimation, applied to independent, short-range dependent and as well long-range dependent sequences.