Wishart and Pseudo-Wishart distributions under elliptical laws and related distributions in the shape theory context

In this paper, we determine the density of a singular elliptically contoured matrix. From this, the study of Wishart and Pseudo-Wishart distributions, whether central or noncentral, whether singular or nonsingular, is extended to the case of elliptical models. Some elated distributions are studied in the context of shape theory. Particular attention is paid to singular size-and-shape and size-and-shape cone densities.