Estimation of autocovariance matrices for high dimensional linear processes
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Estimation of autocovariance matrices for high dimensional linear processes 1 Konrad Furmanczyk ´
Received: 25 January 2019 © The Author(s) 2020
Abstract In this paper under some mild restrictions upper bounds on the rate of convergence for estimators of p × 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 = O n γ /2 for some γ > 1, and in the general case when p 2/β (n −1 log p)1/2 → 0 for some β > 2 as p = p(n) → ∞ and the sample size n → ∞. In particular our results hold for multivariate A R 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. Keywords High dimensional data · Linear process · Autocovariance matrix
1 Introduction Estimation of covariance matrices in a high dimensional setting has been one of the fundamental statistical issues in the last decade. Some statistical applications of estimation of covariance matrices have been presented for ridge regression in Hoerl and Kennard (1970), in regularized discriminant analysis—(Friedman 1989) and in principal component analysis—(Johnstone and Lu 2009). For an overview of this topic and its applications see (Bickel and Levina 2008b; Birnbaum and Nadler 2012; Chen et al. 2013; Fan et al. 2006; Rothman et al. 2009). The problem of estimating covariance matrices for dependent data has been recently investigated by (Chen et al. 2013; Bhattacharjee and Bose 2014a, b; Guo et al. 2016; Jentsch and Politis 2015; McMurry and Politis 2010), and Wu and Pourahmadi (2009). The estimation of the inverse covariance matrix is used in the recovery of the true unknown structure of undirected
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Konrad Furma´nczyk [email protected] Institute of Information Technology, Warsaw University of Life Sciences (SGGW), Nowoursynowska 159, 02-776 Warsaw, Poland
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K. 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. The recovery of undirected graphs on the basis of the estimation of the precision matrices for a general class of nonstationary time series is considered in Xu et al. (2020). Consider a p-dimensional linear process Xt =
∞
j εt− j (almost surely),
(1)
j=0
where the j are p × p matrices, εi = εi,1 , . . . , εi, p , (εt ) are i.i.d. vectors in R p with mean 0 and variance-covariance matrix . Under a causality condition, a vector A R M A process which is a basic model in econometrics and finance is a linear process (Brockwell and Davis 2002). We assume that (εt ) satisfies one of the following conditions: with mean 0 and variance-covariance matrix . (Gauss) εi is Gaussian (SGauss) εi,l ε j,s is sub-Gaussian with constant σ 2 , that is, E exp uεi,l ε j,s ≤ exp σ 2 u 2 /2 for all u ∈ R, i, j = 1, 2, ..
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