Open Access
September, 1991 The Asymptotic Behavior of Some Nonparametric Change-Point Estimators
L. Dumbgen
Ann. Statist. 19(3): 1471-1495 (September, 1991). DOI: 10.1214/aos/1176348257


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.


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L. Dumbgen. "The Asymptotic Behavior of Some Nonparametric Change-Point Estimators." Ann. Statist. 19 (3) 1471 - 1495, September, 1991.


Published: September, 1991
First available in Project Euclid: 12 April 2007

zbMATH: 0776.62032
MathSciNet: MR1126333
Digital Object Identifier: 10.1214/aos/1176348257

Primary: 62G05
Secondary: 62G15

Keywords: bootstrap confidence sets , Change-point , Kolmogorov-Smirnov , Mann-Whitney , nonparametric estimation , random seminorm

Rights: Copyright © 1991 Institute of Mathematical Statistics

Vol.19 • No. 3 • September, 1991
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