Moment for the inverse Riesz distributions

The Riesz distributions dealing with positive definite symmetric matrices are usually used to introduce the class of the inverse Riesz distributions. The latter represents the natural extension of the class of the inverse Wishart. In this paper, we first present a sufficient condition allowing the existence of the expectation of the inverse Riesz distribution. Then, we compute it explicitly. For this purpose, we basically use the Cholesky decomposition as well as an important relation satisfied by the first derivative of continuous Riesz distribution's density. The importance of this first moment consists in the fact that it can be used to estimate the shape parameter through the method of moments.