Open Access
June 2022 Optimal difference-based variance estimators in time series: A general framework
Kin Wai Chan
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Ann. Statist. 50(3): 1376-1400 (June 2022). DOI: 10.1214/21-AOS2154


Variance estimation is important for statistical inference. It becomes nontrivial when observations are masked by serial dependence structures and time-varying mean structures. Existing methods either ignore or sub-optimally handle these nuisance structures. This paper develops a general framework for the estimation of the long-run variance for time series with nonconstant means. The building blocks are difference statistics. The proposed class of estimators is general enough to cover many existing estimators. Necessary and sufficient conditions for consistency are investigated. The first asymptotically optimal estimator is derived. Our proposed estimator is theoretically proven to be invariant to arbitrary mean structures, which may include trends and a possibly divergent number of discontinuities.

Funding Statement

This research was partially supported by grants GRF-2130730 and GRF-2130788 provided by Research Grants Council of HKSAR.


The authors would like to thank the anonymous referees, an associate editor and the editor for their constructive comments that improved the scope and presentation of the paper.


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Kin Wai Chan. "Optimal difference-based variance estimators in time series: A general framework." Ann. Statist. 50 (3) 1376 - 1400, June 2022.


Received: 1 January 2021; Revised: 1 November 2021; Published: June 2022
First available in Project Euclid: 16 June 2022

MathSciNet: MR4441124
zbMATH: 07547934
Digital Object Identifier: 10.1214/21-AOS2154

Primary: 62G05
Secondary: 62G20

Keywords: change point detection , nonlinear time series , optimal bandwidth selection , trend inference , variate difference method

Rights: Copyright © 2022 Institute of Mathematical Statistics

Vol.50 • No. 3 • June 2022
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