A generalization of the Wishart distribution for the elliptical model and its moments for the multivariate t model
We consider the elliptical distribution of n p-dimensional random vectors \(X_ 1,...,X_ n\) having p.d.f. of the form \[ k(n,p)| \Lambda |^{-n/2}g(\sum^{n}_{j=1}(X_ j-\theta)'\Lambda^{-1}(X_ j- \theta)) \] as a generalization of the multivariate normal distribution. Let A denote the Wishart matrix defined by \(A=\sum^{n}_{j=1}(X_ j- \bar X)(X_ j-\bar X)'\), where the vector \(\bar X\) is given by \(\bar X=(1/n)\sum^{n}_{j=1}\) \(X_ j=(\bar X_ 1,...,\bar X_ p)'.\) 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 are computed.
generalization of the multivariate normal distribution
scientific article; zbMATH DE number 4092580