December 2023 Local Whittle estimation of high-dimensional long-run variance and precision matrices
Changryong Baek, Marie-Christine Düker, Vladas Pipiras
Author Affiliations +
Ann. Statist. 51(6): 2386-2414 (December 2023). DOI: 10.1214/23-AOS2330


This work develops nonasymptotic theory for estimation of the long-run variance matrix and its inverse, the so-called precision matrix, for high-dimensional time series under general assumptions on the dependence structure including long-range dependence. The estimation involves shrinkage techniques, which are thresholding and penalizing versions of the classical multivariate local Whittle estimator. The results ensure consistent estimation in a double asymptotic regime where the number of component time series is allowed to grow with the sample size as long as the true model parameters are sparse. The key technical result is a concentration inequality of the local Whittle estimator for the long-run variance matrix around the true model parameters. In particular, it handles simultaneously the estimation of the memory parameters, which enter the underlying model. Novel algorithms for the considered procedures are proposed, and a simulation study and a data application are also provided.

Funding Statement

The first author was supported by the National Research Foundation of Korea (NRF-2019R1F1A1057104). The second author was supported by the DFG (RTG 2131) and NSF Grant 1934985. The third author was supported in part by NSF Grant DMS-1712966.


We would like to thank the Editor, the Associate Editor and anonymous referees for careful reading of our paper and many useful comments and suggestions. This work was carried out during stays of the first and second authors in the Department of Statistics and Operation Research at the University of North Carolina, Chapel Hill. The first and second authors thank the department, in particular, Vladas Pipiras for their hospitality.


Download Citation

Changryong Baek. Marie-Christine Düker. Vladas Pipiras. "Local Whittle estimation of high-dimensional long-run variance and precision matrices." Ann. Statist. 51 (6) 2386 - 2414, December 2023.


Received: 1 December 2022; Revised: 1 June 2023; Published: December 2023
First available in Project Euclid: 20 December 2023

MathSciNet: MR4682702
zbMATH: 07783620
Digital Object Identifier: 10.1214/23-AOS2330

Primary: 62H12 , 62M15
Secondary: 62H20

Keywords: Frequency domain , high-dimensional time series , local Whittle estimation , Short- and long-range dependence , shrinkage estimation , spectral density estimation

Rights: Copyright © 2023 Institute of Mathematical Statistics


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Vol.51 • No. 6 • December 2023
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