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Covariance matrix selection and estimation via penalised normal likelihood. (English) Zbl 1152.62346
Summary: 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 \(L_1\) and \(L_2\) penalties are shown to be closely related to R. Tibshirani’s [J. R. Stat. Soc., Ser. B 58, No. 1, 267–288 (1996)] LASSO approach and to ridge regression. Adding either penalty to the likeliliood helps to produce more stable estimators by introducing shrinkage to the elements in the Cholesky factor, while, because of its singularity, the \(L_1\) 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 simulations and a real data set involving estimation of a \(102\times 102\) covariance matrix.

62H12 Estimation in multivariate analysis
65C60 Computational problems in statistics (MSC2010)
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