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High‐dimensional covariance matrix estimation using a low‐rank and diagonal decomposition
Author(s) -
Wu Yilei,
Qin Yingli,
Zhu Mu
Publication year - 2020
Publication title -
canadian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.804
H-Index - 51
eISSN - 1708-945X
pISSN - 0319-5724
DOI - 10.1002/cjs.11532
Subject(s) - mathematics , covariance matrix , estimator , estimation of covariance matrices , covariance , rank (graph theory) , covariance function , consistency (knowledge bases) , mathematical optimization , statistics , algorithm , combinatorics , geometry
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