Abstract
In this paper under some mild restrictions upper bounds on the rate of convergence for estimators of ptimes p autocovariance and precision matrices for high dimensional linear processes are given. We show that these estimators are consistent in the operator norm in the sub-Gaussian case when p={mathcal {O}}left( n^{gamma /2}right) for some gamma >1, and in the general case when p^{2/beta }(n^{-1} log p)^{1/2}rightarrow 0 for some beta >2 as p=p(n)rightarrow infty and the sample size nrightarrow infty . In particular our results hold for multivariate AR processes. We compare our results with those previously obtained in the literature for independent and dependent data. We also present non-asymptotic bounds for the error probability of these estimators.
Highlights
Estimation of covariance matrices in a high dimensional setting has been one of the fundamental statistical issues in the last decade
The estimation of the inverse covariance matrix is used in the recovery of the true unknown structure of undirected
Furmanczyk graphical models, especially in Gaussian graphical models, where a zero entry of the inverse covariance matrix is associated with a missing edge between two vertices in the graph
Summary
Estimation of covariance matrices in a high dimensional setting has been one of the fundamental statistical issues in the last decade. Bhattacharjee and Bose (2014a) considered the estimation of the high dimensional variance-covariance matrix under a general Gaussian model with weak dependence in both rows and columns of the data matrix. They showed that the bounded and tapered sample variance-covariance matrices are consistent under a suitable column dependence model. In “Comparison of our results with previous studies” section we compare our results with the results obtained by (Bickel and Levina 2008a, b) for independent normal and nonnormal data, with the minimax upper bound for tapering estimator in Cai et al (2010) and with the results for dependent data obtained by (Chen et al 2013; Bhattacharjee and Bose 2014b; Guo et al 2016) and Jentsch and Politis (2015). All the proofs and auxiliary lemmas are given in the “Appendix”
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