The Annals of Statistics

Nonparametric maximum likelihood approach to multiple change-point problems

Changliang Zou, Guosheng Yin, Long Feng, and Zhaojun Wang

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In multiple change-point problems, different data segments often follow different distributions, for which the changes may occur in the mean, scale or the entire distribution from one segment to another. Without the need to know the number of change-points in advance, we propose a nonparametric maximum likelihood approach to detecting multiple change-points. Our method does not impose any parametric assumption on the underlying distributions of the data sequence, which is thus suitable for detection of any changes in the distributions. The number of change-points is determined by the Bayesian information criterion and the locations of the change-points can be estimated via the dynamic programming algorithm and the use of the intrinsic order structure of the likelihood function. Under some mild conditions, we show that the new method provides consistent estimation with an optimal rate. We also suggest a prescreening procedure to exclude most of the irrelevant points prior to the implementation of the nonparametric likelihood method. Simulation studies show that the proposed method has satisfactory performance of identifying multiple change-points in terms of estimation accuracy and computation time.

Article information

Ann. Statist., Volume 42, Number 3 (2014), 970-1002.

First available in Project Euclid: 20 May 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G05: Estimation
Secondary: 62G20: Asymptotic properties

BIC change-point estimation Cramér–von Mises statistic dynamic programming empirical distribution function goodness-of-fit test


Zou, Changliang; Yin, Guosheng; Feng, Long; Wang, Zhaojun. Nonparametric maximum likelihood approach to multiple change-point problems. Ann. Statist. 42 (2014), no. 3, 970--1002. doi:10.1214/14-AOS1210.

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Supplemental materials

  • Supplementary material: Supplement to “Nonparametric maximum likelihood approach to multiple change-point problems”. We provide technical details for the proof of Corollary 1, and additional simulation results.