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

John Barnard, Robert McCulloch, Xiao Li Meng

Research output: Contribution to journalArticle

297 Citations (Scopus)

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 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.

Original languageEnglish (US)
Pages (from-to)1281-1311
Number of pages31
JournalStatistica Sinica
Volume10
Issue number4
StatePublished - Oct 2000
Externally publishedYes

Fingerprint

Shrinkage
Standard deviation
Covariance matrix
Shrinkage Estimation
Modeling
Correlation Matrix
Location-scale Model
Cholesky Decomposition
Decompose
Categorical variable
Spectral Decomposition
Variance Components
Continuous Variables
Regression Coefficient
Statistical Inference
Logarithm
Dimensionality
Non-negative
Model
Decomposition

Keywords

  • General location model
  • General location-scale model
  • Gibbs sampler
  • Hierarchical models
  • Markov chain Monte Carlo
  • Wishart distribution

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Cite this

Modeling covariance matrices in terms of standard deviations and correlations, with application to shrinkage. / Barnard, John; McCulloch, Robert; Meng, Xiao Li.

In: Statistica Sinica, Vol. 10, No. 4, 10.2000, p. 1281-1311.

Research output: Contribution to journalArticle

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