## The matrix-F prior for estimating and testing covariance matrices.(English)Zbl 1407.62080

Summary: 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.

### MSC:

 62F15 Bayesian inference 62H12 Estimation in multivariate analysis 62H10 Multivariate distribution of statistics 62J12 Generalized linear models (logistic models)

BayesDA; BOCOR
Full Text:

### References:

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