Inverse Wishart distributions based on singular elliptically contoured distribution

Assuming that the random matrix X has a singular or non-singular matrix variate elliptically contoured distribution, the density function of the Moore–Penrose inverse Z=(X′X)+ is given with respect to the Hausdorff measure. The result is applied to Bayesian inference for a general multivariate linear regression model with matrix variate elliptically distributed errors. Some results concerning the posterior joint and marginal distributions of the parameters are obtained.