Modeling Covariance Matrices in Terms of Standard Deviations and Correlations, with Application to Shrinkage

The covariance matrix plays an important role in statistical inference, yet modeling a covariance matrix is often a difficult task in practice due to its dimensionality and the non-negative definite constraint. In order to model a covariance matrix effectively, it is typically broken down into components based on modeling considerations or mathematical convenience. Decompositions that have received recent research attention include variance components, spectral decomposition, Cholesky decomposition, and matrix logarithm. In this paper we study a statistically motivated decomposition which appears to be relatively unexplored for the purpose of modeling. We model a covariance matrix in terms of its corresponding standard deviations and correlation matrix. We discuss two general modeling situations where this approach is useful: shrinkage estimation of regression coefficients, and a general location-scale model for both categorical and continuous variables. We present some simple choices for priors in terms of standard deviations and the correlation matrix, and describe a straightforward computational strategy for obtaining the posterior of the covariance matrix. We apply our method to real and simulated data sets in the context of shrinkage estimation.