Abstract: The canonical parameter of a covariance selection model is the inverse covariance matrix Σ -1 whose zero pattern gives the conditional independence structure characterising the model. In this paper we consider the upper triangular matrix Φ obtained by the Cholesky decomposition Σ - 1 = Φ T Φ. This provides an interesting alternative parameterisation of decomposable models since its upper triangle has the same zero structure as Σ -1 and its elements have an interpretation as parameters of certain conditional distributions. For a distribution for Σ, the strong hyper-Markov property is shown to be characterised by the mutual independence of the rows of Φ. This is further used to generalise to the hyper inverse Wishart distribution some well-known properties of the inverse Wishart distribution. In particular we show that a hyper inverse Wishart matrix can be decomposed into independent normal and chi-squared random variables, and we describe a family of transformations under which the family of hyper inverse Wishart distributions is closed.

Abstract: The calculation of moments of complex Wishart and complex inverse Wishart distributed random matrices is addressed. In applications such as radar, sonar or seismics, complex Wishart and complex inverse Wishart distributed random matrices are used to model the statistical properties of complex sample covariance matrices and complex inverse sample covariance matrices, respectively. Moments of these random matrices are often needed, for example, in studies of the asymptotic properties of parameter estimates. A derivation of the probability density function of complex inverse Wishart distributed random matrices is given. Furthermore, strategies are outlined for the calculation of the moments of both complex Wishart and complex inverse Wishart distributed matrices.