The Annals of Statistics

Estimation of large covariance and precision matrices from temporally dependent observations

Hai Shu and Bin Nan

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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.

Article information

Ann. Statist., Volume 47, Number 3 (2019), 1321-1350.

Received: July 2017
Revised: December 2017
First available in Project Euclid: 13 February 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H12: Estimation
Secondary: 62H35: Image analysis

Brain functional connectivity correlation matrix heavy tail high-dimensional data long memory minimax optimal convergence rates nonstationarity sub-Gaussian tail temporal dependence


Shu, Hai; Nan, Bin. Estimation of large covariance and precision matrices from temporally dependent observations. Ann. Statist. 47 (2019), no. 3, 1321--1350. doi:10.1214/18-AOS1716.

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Supplemental materials

  • Supplement to “Estimation of large covariance and precision matrices from temporally dependent observations”. The Supplementary Material contains technical preparations, detailed proofs of the technical lemmas given in the Appendix and all the theorems in the main text, useful numerical considerations and additional results of the rfMRI data analysis.