The Annals of Statistics

Change-point in stochastic design regression and the bootstrap

Emilio Seijo and Bodhisattva Sen

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In this paper we study the consistency of different bootstrap procedures for constructing confidence intervals (CIs) for the unique jump discontinuity (change-point) in an otherwise smooth regression function in a stochastic design setting. This problem exhibits nonstandard asymptotics, and we argue that the standard bootstrap procedures in regression fail to provide valid confidence intervals for the change-point. We propose a version of smoothed bootstrap, illustrate its remarkable finite sample performance in our simulation study and prove the consistency of the procedure. The m out of n bootstrap procedure is also considered and shown to be consistent. We also provide sufficient conditions for any bootstrap procedure to be consistent in this scenario.

Article information

Ann. Statist., Volume 39, Number 3 (2011), 1580-1607.

First available in Project Euclid: 7 June 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G08: Nonparametric regression 62G05: Estimation
Secondary: 62G20: Asymptotic properties

Argmax continuous mapping theorem consistency of the bootstrap m out of n bootstrap nonstandard asymptotics semiparametric regression smoothed bootstrap


Seijo, Emilio; Sen, Bodhisattva. Change-point in stochastic design regression and the bootstrap. Ann. Statist. 39 (2011), no. 3, 1580--1607. doi:10.1214/11-AOS874.

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Supplemental materials

  • Supplementary material: Supplement to “Change-point in stochastic design regression and the bootstrap”. The supplementary file contains a longer version of this paper with all the technical details which were excluded in the present version due to their length.