ABSTRACT
No general representations of the density or the distribution function of quadratic forms in elliptically contoured vectors are currently available in the literature. As a result, the random vectors associated with quadratic forms have been assumed—almost invariably and at times inappropriately—to follow a multivariate normal distribution in statistical applications. It is shown in this paper that a certain scale mixture representation of the density function of a central elliptically contoured vector, as well as its extension to non-central random vectors yield computable expressions for the moments, the density and the distribution function of quadratic forms in elliptically contoured vectors. The same approach can readily be used to determine the distribution of other functions of elliptically contoured vectors in terms of their Gaussian counterparts.
