### 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 language | English (US) |
---|---|

Pages (from-to) | 1281-1311 |

Number of pages | 31 |

Journal | Statistica Sinica |

Volume | 10 |

Issue number | 4 |

State | Published - Oct 2000 |

Externally published | Yes |

### Fingerprint

### 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

*Statistica Sinica*,

*10*(4), 1281-1311.

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

Research output: Contribution to journal › Article

*Statistica Sinica*, vol. 10, no. 4, pp. 1281-1311.

}

TY - JOUR

T1 - Modeling covariance matrices in terms of standard deviations and correlations, with application to shrinkage

AU - Barnard, John

AU - McCulloch, Robert

AU - Meng, Xiao Li

PY - 2000/10

Y1 - 2000/10

N2 - 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.

AB - 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.

KW - General location model

KW - General location-scale model

KW - Gibbs sampler

KW - Hierarchical models

KW - Markov chain Monte Carlo

KW - Wishart distribution

UR - http://www.scopus.com/inward/record.url?scp=0034557223&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0034557223&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0034557223

VL - 10

SP - 1281

EP - 1311

JO - Statistica Sinica

JF - Statistica Sinica

SN - 1017-0405

IS - 4

ER -