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Univariate mean change point detection : penalization, CUSUM and optimality
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Wang, Daren, Yu, Yi and Rinaldo, Alessandro (2020) Univariate mean change point detection : penalization, CUSUM and optimality. Electronic Journal of Statistics, 14 (1). pp. 1917-1961. doi:10.1214/20-EJS1710 ISSN 1935-7524.
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Official URL: https://doi.org/10.1214/20-EJS1710
Abstract
The problem of univariate mean change point detection and localization based on a sequence of n independent observations with piecewise constant means has been intensively studied for more than half century, and serves as a blueprint for change point problems in more complex settings. We provide a complete characterization of this classical problem in a general framework in which the upper bound σ2 on the noise variance, the minimal spacing Δ between two consecutive change points and the minimal magnitude κ of the changes, are allowed to vary with n. We first show that consistent localization of the change points, when the signal-to-noise ratio κΔ√σ<log(n)−−−−−√, is impossible. In contrast, when κΔ√σ diverges with n at the rate of at least log(n)−−−−−√, we demonstrate that two computationally-efficient change point estimators, one based on the solution to an ℓ0-penalized least squares problem and the other on the popular wild binary segmentation algorithm, are both consistent and achieve a localization rate of the order σ2κ2log(n). We further show that such rate is minimax optimal, up to a log(n) term.
| Item Type: | Journal Article | ||||||
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| Subjects: | Q Science > QA Mathematics | ||||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Statistics | ||||||
| Library of Congress Subject Headings (LCSH): | Analysis of variance, Mathematical statistics, Change-point problems , CUSUM technique, Binary system (Mathematics) | ||||||
| Journal or Publication Title: | Electronic Journal of Statistics | ||||||
| Publisher: | Institute of Mathematical Statistics | ||||||
| ISSN: | 1935-7524 | ||||||
| Official Date: | 28 April 2020 | ||||||
| Dates: |
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| Volume: | 14 | ||||||
| Number: | 1 | ||||||
| Page Range: | pp. 1917-1961 | ||||||
| DOI: | 10.1214/20-EJS1710 | ||||||
| Status: | Peer Reviewed | ||||||
| Publication Status: | Published | ||||||
| Access rights to Published version: | Open Access (Creative Commons) | ||||||
| Date of first compliant deposit: | 2 April 2020 | ||||||
| Date of first compliant Open Access: | 28 April 2020 | ||||||
| Related URLs: | |||||||
| Open Access Version: |
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