Open Access
August 2020 Estimation and inference for precision matrices of nonstationary time series
Xiucai Ding, Zhou Zhou
Ann. Statist. 48(4): 2455-2477 (August 2020). DOI: 10.1214/19-AOS1894


We consider the estimation of and inference on precision matrices of a rich class of univariate locally stationary linear and nonlinear time series, assuming that only one realization of the time series is observed. Using a Cholesky decomposition technique, we show that the precision matrices can be directly estimated via a series of least squares linear regressions with smoothly time-varying coefficients. The method of sieves is utilized for the estimation and is shown to be optimally adaptive in terms of estimation accuracy and efficient in terms of computational complexity. We establish an asymptotic theory for a class of $\mathcal{L}^{2}$ tests based on the nonparametric sieve estimators. The latter are used for testing whether the precision matrices are diagonal or banded. A Gaussian approximation result is established for a wide class of quadratic forms of nonstationary and possibly nonlinear processes of diverging dimensions which is of interest by itself.


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Xiucai Ding. Zhou Zhou. "Estimation and inference for precision matrices of nonstationary time series." Ann. Statist. 48 (4) 2455 - 2477, August 2020.


Received: 1 March 2019; Revised: 1 August 2019; Published: August 2020
First available in Project Euclid: 14 August 2020

MathSciNet: MR4134802
Digital Object Identifier: 10.1214/19-AOS1894

Primary: 62G05 , 62G10 , 62M10

Keywords: Cholesky decomposition , high-dimensional Gaussian approximation , nonstationary time series , precision matrices , random matrices , sieve estimation , white noise and bandedness tests

Rights: Copyright © 2020 Institute of Mathematical Statistics

Vol.48 • No. 4 • August 2020
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