On the Exact Distribution of the Product of an Inverse Wishart Matrix and a Normal Vector

We derive the exact density function of the product of an inverseWishart random matrix and an independent normal random vector. The density is expressed as a one-dimensional integral that can be evaluated quickly and accurately by standard quadrature methods. We further obtain integral representations for the density of linear combinations of the elements of this product; in particular, the joint density of any p linear combinations can be recovered from an integral of dimension at most min(p+1, m− p+1, 6), where m is the dimension of the random vector. All results are established in the general setting in which the scale matrix of the Wishart distribution and the covariance matrix of the normal vector are arbitrary positive definite matrices, and simplified formulas involving integrals of dimension at most three are provided for the important special case in which the two covariance matrices are proportional.