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Flexible covariance estimation in graphical Gaussian models. (English) Zbl 1168.62054
Summary: We propose a class of Bayes estimators for the covariance matrix of graphical Gaussian models being Markov with respect to a decomposable graph \(G\). Working with the \(W_{P_G}\) family, defined by G. Letac and H. Massam [Ann. Stat. 35, No. 3, 1278–1323 (2007; Zbl 1194.62078)], we derive closed-form expressions for Bayes estimators under the entropy and squared-error losses. The \(W_{P_G}\) family includes the classical inverse of the hyper inverse Wishart but has many more shape parameters, thus allowing for flexibility in differentially shrinking various parts of the covariance matrix. Moreover, using this family avoids recourse to MCMC, often infeasible in high-dimensional problems. We illustrate the performance of our estimators through a collection of numerical examples where we explore frequentist risk properties and the efficacy of graphs in the estimation of high-dimensional covariance structures.

MSC:
62H12 Estimation in multivariate analysis
62C10 Bayesian problems; characterization of Bayes procedures
05C90 Applications of graph theory
62F15 Bayesian inference
Software:
HdBCS
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References:
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