Wishart and pseudo-Wishart distributions and some applications to shape theory

Suppose that X similar to N-N x m(mu, Sigma, Theta). An expression for the density function is given when Sigma greater than or equal to 0 and/or Theta greater than or equal to 0. An extension of Uhlig's result (Uhlig [17]) is expanded for the singular value decomposition of a matrix Z of order N x m when the rank (Z) = q less than or equal to min(N, m). This paper fills an important gap in unifying, for the first time, all Wishart and pseudo-Wishart distributions, whether central or noncentral, whether singular or nonsingular, and applying them in shape analysis. In particular, the shape density and the size-and-shape cone density are obtained for the singular general case. (C) 1997 Academic Press.