## The Annals of Statistics

### Covariance and precision matrix estimation for high-dimensional time series

#### Abstract

We consider estimation of covariance matrices and their inverses (a.k.a. precision matrices) for high-dimensional stationary and locally stationary time series. In the latter case the covariance matrices evolve smoothly in time, thus forming a covariance matrix function. Using the functional dependence measure of Wu [Proc. Natl. Acad. Sci. USA 102 (2005) 14150–14154 (electronic)], we obtain the rate of convergence for the thresholded estimate and illustrate how the dependence affects the rate of convergence. Asymptotic properties are also obtained for the precision matrix estimate which is based on the graphical Lasso principle. Our theory substantially generalizes earlier ones by allowing dependence, by allowing nonstationarity and by relaxing the associated moment conditions.

#### Article information

Source
Ann. Statist., Volume 41, Number 6 (2013), 2994-3021.

Dates
First available in Project Euclid: 1 January 2014

https://projecteuclid.org/euclid.aos/1388545676

Digital Object Identifier
doi:10.1214/13-AOS1182

Mathematical Reviews number (MathSciNet)
MR3161455

Zentralblatt MATH identifier
1294.62123

Subjects
Primary: 62H12: Estimation

#### Citation

Chen, Xiaohui; Xu, Mengyu; Wu, Wei Biao. Covariance and precision matrix estimation for high-dimensional time series. Ann. Statist. 41 (2013), no. 6, 2994--3021. doi:10.1214/13-AOS1182. https://projecteuclid.org/euclid.aos/1388545676

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