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High-dimensional covariance matrix estimation using a low-rank and diagonal decomposition
Authors:Yilei Wu  Yingli Qin  Mu Zhu
Institution:Department of Statistics and Actuarial Science, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, Canada, N2L 3G1
Abstract:We study high-dimensional covariance/precision matrix estimation under the assumption that the covariance/precision matrix can be decomposed into a low-rank component L and a diagonal component D. The rank of L can either be chosen to be small or controlled by a penalty function. Under moderate conditions on the population covariance/precision matrix itself and on the penalty function, we prove some consistency results for our estimators. A block-wise coordinate descent algorithm, which iteratively updates L and D, is then proposed to obtain the estimator in practice. Finally, various numerical experiments are presented; using simulated data, we show that our estimator performs quite well in terms of the Kullback–Leibler loss; using stock return data, we show that our method can be applied to obtain enhanced solutions to the Markowitz portfolio selection problem. The Canadian Journal of Statistics 48: 308–337; 2020 © 2019 Statistical Society of Canada
Keywords:Akaike information criterion  consistency  coordinate descent  eigen-decomposition  Kullback–Leibler loss  log-determinant semi-definite programming  Markowitz portfolio selection
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