Electronic Journal of Statistics

Univariate mean change point detection: Penalization, CUSUM and optimality

Daren Wang, Yi Yu, and Alessandro Rinaldo

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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 $\sigma ^{2}$ on the noise variance, the minimal spacing $\Delta $ between two consecutive change points and the minimal magnitude $\kappa $ of the changes, are allowed to vary with $n$. We first show that consistent localization of the change points is impossible in the low signal-to-noise ratio regime $\frac{\kappa \sqrt{\Delta }}{\sigma }\preceq \sqrt{\log (n)}$. In contrast, when $\frac{\kappa \sqrt{\Delta }}{\sigma }$ diverges with $n$ at the rate of at least $\sqrt{\log (n)}$, we demonstrate that two computationally-efficient change point estimators, one based on the solution to an $\ell _{0}$-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 $\frac{\sigma ^{2}}{\kappa ^{2}}\log (n)$. We further show that such rate is minimax optimal, up to a $\log (n)$ term.

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Electron. J. Statist., Volume 14, Number 1 (2020), 1917-1961.

Received: June 2019
First available in Project Euclid: 28 April 2020

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Change point detection minimax optimality $\ell _{0}$-penalization CUSUM statistics binary segmentation

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Wang, Daren; Yu, Yi; Rinaldo, Alessandro. Univariate mean change point detection: Penalization, CUSUM and optimality. Electron. J. Statist. 14 (2020), no. 1, 1917--1961. doi:10.1214/20-EJS1710. https://projecteuclid.org/euclid.ejs/1588039326

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