Modeling covariance matrices in terms of standard deviations and correlations, with application to shrinkage

Abstract

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 co- variance 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 decompo- sition, 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 correspond- ing standard deviations and correlation matrix. We discuss two general modeling situations where this approach is useful: shrinkage estimation of regression co- efficients, 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.