## Bayesian Analysis

### The Matrix-$F$ Prior for Estimating and Testing Covariance Matrices

#### Abstract

The matrix-$F$ distribution is presented as prior for covariance matrices as an alternative to the conjugate inverted Wishart distribution. A special case of the univariate $F$ distribution for a variance parameter is equivalent to a half-$t$ distribution for a standard deviation, which is becoming increasingly popular in the Bayesian literature. The matrix-$F$ distribution can be conveniently modeled as a Wishart mixture of Wishart or inverse Wishart distributions, which allows straightforward implementation in a Gibbs sampler. By mixing the covariance matrix of a multivariate normal distribution with a matrix-$F$ distribution, a multivariate horseshoe type prior is obtained which is useful for modeling sparse signals. Furthermore, it is shown that the intrinsic prior for testing covariance matrices in non-hierarchical models has a matrix-$F$ distribution. This intrinsic prior is also useful for testing inequality constrained hypotheses on variances. Finally through simulation it is shown that the matrix-variate $F$ distribution has good frequentist properties as prior for the random effects covariance matrix in generalized linear mixed models.

#### Article information

Source
Bayesian Anal., Volume 13, Number 4 (2018), 1193-1214.

Dates
First available in Project Euclid: 12 January 2018

https://projecteuclid.org/euclid.ba/1515747744

Digital Object Identifier
doi:10.1214/17-BA1092

Mathematical Reviews number (MathSciNet)
MR3855368

#### Citation

Mulder, Joris; Pericchi, Luis Raúl. The Matrix- $F$ Prior for Estimating and Testing Covariance Matrices. Bayesian Anal. 13 (2018), no. 4, 1193--1214. doi:10.1214/17-BA1092. https://projecteuclid.org/euclid.ba/1515747744

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#### Supplemental materials

• Supplementary material for “The matrix-F prior for estimating and testing covariance matrices”. The Supplementary Material for “The matrix-F prior for estimating and testing covariance matrices” contains a proof that the matrix-F distribution has the reciprocity property (Section 1); a derivation of the means and (co)variances of the elements of a random matrix having a matrix-F distribution (Section 2); the derivation of the intrinsic prior for a precise hypothesis test of a covariance matrix and the resulting intrinsic Bayes factor (Section 3); a proof that the intrinsic Bayes factor is consistent (Section 4); and a derivation of the Bayes factor of an inequality-constrained covariance matrix against an unconstrained covariance matrix (Section 5).