Open Access
2010 Adaptive estimation of covariance matrices via Cholesky decomposition
Nicolas Verzelen
Electron. J. Statist. 4: 1113-1150 (2010). DOI: 10.1214/10-EJS580


This paper studies the estimation of a large covariance matrix. We introduce a novel procedure called ChoSelect based on the Cholesky factor of the inverse covariance. This method uses a dimension reduction strategy by selecting the pattern of zero of the Cholesky factor. Alternatively, ChoSelect can be interpreted as a graph estimation procedure for directed Gaussian graphical models. Our approach is particularly relevant when the variables under study have a natural ordering (e.g. time series) or more generally when the Cholesky factor is approximately sparse. ChoSelect achieves non-asymptotic oracle inequalities with respect to the Kullback-Leibler entropy. Moreover, it satisfies various adaptive properties from a minimax point of view. We also introduce and study a two-stage procedure that combines ChoSelect with the Lasso. This last method enables the practitioner to choose his own trade-off between statistical efficiency and computational complexity. Moreover, it is consistent under weaker assumptions than the Lasso. The practical performances of the different procedures are assessed on numerical examples.


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Nicolas Verzelen. "Adaptive estimation of covariance matrices via Cholesky decomposition." Electron. J. Statist. 4 1113 - 1150, 2010.


Published: 2010
First available in Project Euclid: 28 October 2010

zbMATH: 1329.62265
MathSciNet: MR2735882
Digital Object Identifier: 10.1214/10-EJS580

Primary: 62H12
Secondary: 62F35 , 62J05

Keywords: banding , Cholesky decomposition , Covariance matrix , directed graphical models , minimax rate of estimation , penalized criterion

Rights: Copyright © 2010 The Institute of Mathematical Statistics and the Bernoulli Society

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