A generalization of the Wishart distribution for the elliptical model and its moments for the multivariate t model

Abstract

We consider the elliptical distribution of n p-dimensional random vectors X 1, …, X n having p.d.f. of the form k(n, p) |Λ| −n 2 g(Σ j=1 n(X j−θ)′ Λ −1(X j−θ)) as a generalization of the multivariate normal distribution. Let A denote the Wishart matrix defined by A = Σ j=1 n(X j− X)(X j− X)′ , where the vector X is given by X = ( 1 n ) Σ j=1 n X j = ( X 1, …, X p)′ . In this paper we derive the distribution of A when X 1, …, X n is assumed to have an elliptical distribution. This result is specialized to the case where X 1, …, X n is assumed to have a multivariate t distribution, a subclass of the elliptical class of distributions. Furthermore, the first two moments of A for this subclass is computed.