The Annals of Statistics

Gradient-based structural change detection for nonstationary time series M-estimation

Weichi Wu and Zhou Zhou

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We consider structural change testing for a wide class of time series M-estimation with nonstationary predictors and errors. Flexible predictor-error relationships, including exogenous, state-heteroscedastic and autoregressive regressions and their mixtures, are allowed. New uniform Bahadur representations are established with nearly optimal approximation rates. A CUSUM-type test statistic based on the gradient vectors of the regression is considered. In this paper, a simple bootstrap method is proposed and is proved to be consistent for M-estimation structural change detection under both abrupt and smooth nonstationarity and temporal dependence. Our bootstrap procedure is shown to have certain asymptotically optimal properties in terms of accuracy and power. A public health time series dataset is used to illustrate our methodology, and asymmetry of structural changes in high and low quantiles is found.

Article information

Ann. Statist., Volume 46, Number 3 (2018), 1197-1224.

Received: August 2016
Revised: February 2017
First available in Project Euclid: 3 May 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62J20: Diagnostics 62M10: Time series, auto-correlation, regression, etc. [See also 91B84] 62G09: Resampling methods 62G10: Hypothesis testing

M-estimation piecewise local stationarity bootstrap structural change


Wu, Weichi; Zhou, Zhou. Gradient-based structural change detection for nonstationary time series M-estimation. Ann. Statist. 46 (2018), no. 3, 1197--1224. doi:10.1214/17-AOS1582.

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Supplemental materials

  • Supplement to “Gradient-based structural change detection for nonstationary time series M-estimation”. We provide (a) detailed proofs of the theorems and lemmas, (b) theoretical investigation on parameter estimation under the alternative hypothesis, (c) the analysis of dynamic models, (d) extension of our methodology to finitely many M-estimations and (e) extra simulation results.