Open Access
June 2019 Estimation of large covariance and precision matrices from temporally dependent observations
Hai Shu, Bin Nan
Ann. Statist. 47(3): 1321-1350 (June 2019). DOI: 10.1214/18-AOS1716


We consider the estimation of large covariance and precision matrices from high-dimensional sub-Gaussian or heavier-tailed observations with slowly decaying temporal dependence. The temporal dependence is allowed to be long-range so with longer memory than those considered in the current literature. We show that several commonly used methods for independent observations can be applied to the temporally dependent data. In particular, the rates of convergence are obtained for the generalized thresholding estimation of covariance and correlation matrices, and for the constrained $\ell_{1}$ minimization and the $\ell_{1}$ penalized likelihood estimation of precision matrix. Properties of sparsistency and sign-consistency are also established. A gap-block cross-validation method is proposed for the tuning parameter selection, which performs well in simulations. As a motivating example, we study the brain functional connectivity using resting-state fMRI time series data with long-range temporal dependence.


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Hai Shu. Bin Nan. "Estimation of large covariance and precision matrices from temporally dependent observations." Ann. Statist. 47 (3) 1321 - 1350, June 2019.


Received: 1 July 2017; Revised: 1 December 2017; Published: June 2019
First available in Project Euclid: 13 February 2019

zbMATH: 07053510
MathSciNet: MR3911114
Digital Object Identifier: 10.1214/18-AOS1716

Primary: 62H12
Secondary: 62H35

Keywords: Brain functional connectivity , Correlation matrix , heavy tail , High-dimensional data , long memory , minimax optimal convergence rates , nonstationarity , sub-Gaussian tail , temporal dependence

Rights: Copyright © 2019 Institute of Mathematical Statistics

Vol.47 • No. 3 • June 2019
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