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Biometrika 2006 93(1):85-98; doi:10.1093/biomet/93.1.85
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© 2006 Biometrika Trust

Covariance matrix selection and estimation via penalised normal likelihood

Jianhua Z. Huang1, Naiping Liu2, Mohsen Pourahmadi3 and Linxu Liu4

1 Department of Statistics, Texas A&M University, College Station, Texas 77843-3143, U.S.A. jianhua{at}stat.tamu.edu, 2 Department of Statistics, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6340, U.S.A. nliu{at}wharton.upenn.edu, 3 Division of Statistics, Northern Illinois University, DeKalb, Illinois 60115-2854, U.S.A. pourahm{at}math.niu.edu, 4 Department of Biostatistics, Mailman School of Public Health, Columbia University, New York, New York 10032, U.S.A. lxliu{at}biostat.columbia.edu

We propose a nonparametric method for identifying parsimony and for producing a statistically efficient estimator of a large covariance matrix. We reparameterise a covariance matrix through the modified Cholesky decomposition of its inverse or the one-step-ahead predictive representation of the vector of responses and reduce the nonintuitive task of modelling covariance matrices to the familiar task of model selection and estimation for a sequence of regression models. The Cholesky factor containing these regression coefficients is likely to have many off-diagonal elements that are zero or close to zero. Penalised normal likelihoods in this situation with L1 and L2 penalities are shown to be closely related to Tibshirani's (1996) LASSO approach and to ridge regression. Adding either penalty to the likelihood helps to produce more stable estimators by introducing shrinkage to the elements in the Cholesky factor, while, because of its singularity, the L1 penalty will set some elements to zero and produce interpretable models. An algorithm is developed for computing the estimator and selecting the tuning parameter. The proposed maximum penalised likelihood estimator is illustrated using simulation and a real dataset involving estimation of a 102 x 102 covariance matrix.

Key Words: Cholesky decomposition; Crossvalidation; LASSO; Lp penalty; Model selection; Penalised likelihood; Shrinkage

Received August 2004. Revised October 2005.


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