### 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.

Original language | English |
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Pages (from-to) | 1193-1214 |

Journal | Bayesian Analysis |

Volume | 13 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2018 |

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## Cite this

Mulder, J., & Pericchi, L. R. (2018). The matrix-F prior for estimating and testing covariance matrices.

*Bayesian Analysis*,*13*(4), 1193-1214. https://doi.org/10.1214/17-BA1092