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

TL;DR

This paper introduces a unified nonparametric change-point estimator that achieves optimal convergence rates, including the classic 1/n rate, across diverse dependent and nonstationary data sequences.

Contribution

It develops a general class of empirical measure-based estimators with proven consistency and optimal convergence rates for nonstationary sequences.

Findings

Achieves 1/n convergence rate for a broad class of processes.

Unifies and extends existing change-point estimation methods.

Effective for independent, short-range, and long-range dependent sequences.

Abstract

Let $(X_i)_{i=1,...,n}$ 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 $i>n\theta$, 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 $\mathcal{F}$. 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.