Regularized estimation of linear functionals of precision matrices for high-dimensional time series

Abstract

This paper studies a Dantzig-selector type regularized estimator for linear functionals of high-dimensional linear processes. Explicit rates of convergence of the proposed estimator are obtained and they cover the broad regime from i.i.d. samples to long-range dependent time series and from sub-Gaussian innovations to those with mild polynomial moments. It is shown that the convergence rates depend on the degree of temporal dependence and the moment conditions of the underlying linear processes. The Dantzig-selector estimator is applied to the sparse Markowitz portfolio allocation and the optimal linear prediction for time series, in which the ratio consistency when compared with an oracle estimator is established. The effect of dependence and innovation moment conditions is further illustrated in the simulation study. Finally, the regularized estimator is applied to classify the cognitive states on a real fMRI dataset and to portfolio optimization on a financial dataset.

44 pages, 4 figures

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• Regularized estimation of large covariance matrices
• Covariance regularization by thresholding
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• Network exploration via the adaptive LASSO and SCAD penalties
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• Sparse linear discriminant analysis by thresholding for high dimensional data
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• Covariance and precision matrix estimation for high-dimensional time series
• L0 Sparse Inverse Covariance Estimation

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• Optimal estimation of functionals of high-dimensional mean and covariance matrix
• Inference of high-dimensional linear models with time-varying coefficients
• A General Theory for Large-Scale Curve Time Series via Functional Stability Measure
• A Bernstein-type Inequality for High Dimensional Linear Processes with Applications to Robust Estimation of Time Series Regressions