The Asymptotic Behavior of Some Nonparametric Change-Point Estimators

Abstract

Consider a sequence $X_1, X_2,\ldots, X_n$ of independent random variables, where $X_1, X_2,\ldots, X_{n\theta}$ have distribution $P,$ and $X_{n\theta + 1}, X_{n\theta + 2},\ldots, X_n$ have distribution $Q$. The change-point $\theta \in (0,1)$ is an unknown parameter to be estimated, and $P$ and $Q$ are two unknown probability distributions. The nonparametric estimators of Darkhovskh and Carlstein are imbedded in a more general framework, where random seminorms are applied to empirical measures for making inference about $\theta$. Carlstein's and Darkhovskh's results about consistency are improved, and the limiting distributions of some particular estimators are derived in various models. Further we propose asymptotically valid confidence regions for the change point $\theta$ by inverting bootstrap tests. As an example this method is applied to the Nile data.