Thresholding and graphical local Whittle estimation

This work develops non-asymptotic theory for estimation of the long-run variance matrix and its inverse, the so-called precision matrix, for high-dimensional Gaussian time series under general assumptions on the dependence structure including long-range dependence. The estimation involves shrinkage techniques which are thresholding and penalizing versions of the classical multivariate local Whittle estimator. The results ensure consistent estimation in a double asymptotic regime where the number of component time series is allowed to grow with the sample size as long as the true model parameters are sparse. The key technical result is a concentration inequality of the local Whittle estimator for the long-run variance matrix around the true model parameters. In particular, it handles simultaneously the estimation of the memory parameters which enter the underlying model. Novel algorithms for the considered procedures are proposed, and a simulation study and a data application are also provided.