The Annals of Statistics

Detecting relevant changes in the mean of nonstationary processes—A mass excess approach

Holger Dette and Weichi Wu

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Abstract

This paper considers the problem of testing if a sequence of means $(\mu_{t})_{t=1,\ldots ,n}$ of a nonstationary time series $(X_{t})_{t=1,\ldots ,n}$ is stable in the sense that the difference of the means $\mu_{1}$ and $\mu_{t}$ between the initial time $t=1$ and any other time is smaller than a given threshold, that is $|\mu_{1}-\mu_{t}|\leq c$ for all $t=1,\ldots ,n$. A test for hypotheses of this type is developed using a bias corrected monotone rearranged local linear estimator and asymptotic normality of the corresponding test statistic is established. As the asymptotic variance depends on the location of the roots of the equation $|\mu_{1}-\mu_{t}|=c$ a new bootstrap procedure is proposed to obtain critical values and its consistency is established. As a consequence we are able to quantitatively describe relevant deviations of a nonstationary sequence from its initial value. The results are illustrated by means of a simulation study and by analyzing data examples.

Article information

Source
Ann. Statist., Volume 47, Number 6 (2019), 3578-3608.

Dates
Received: February 2018
Revised: January 2019
First available in Project Euclid: 31 October 2019

Permanent link to this document
https://projecteuclid.org/euclid.aos/1572487403

Digital Object Identifier
doi:10.1214/19-AOS1811

Mathematical Reviews number (MathSciNet)
MR4025752

Subjects
Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84] 62F05: Asymptotic properties of tests 62G08: Nonparametric regression 62G09: Resampling methods

Keywords
Locally stationary process change point analysis relevant change points local linear estimation Gaussian approximation rearrangement estimators

Citation

Dette, Holger; Wu, Weichi. Detecting relevant changes in the mean of nonstationary processes—A mass excess approach. Ann. Statist. 47 (2019), no. 6, 3578--3608. doi:10.1214/19-AOS1811. https://projecteuclid.org/euclid.aos/1572487403


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Supplemental materials

  • Supplement to “Detecting relevant changes in the mean of nonstationary processes—A mass excess approach”. We provide proof for (B.17) and (B.18) for Theorem 4.1, proof of Theorem 4.4 and technical lemmas in the Supplementary Material.