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

Abstract

Suppose that X ∼ script NN × m(μ, Σ, Θ). An expression for the density function is given when Σ ≥ 0 and/or Θ ≥ 0. An extension of Uhlig's result (Uhlig [17]) is expanded for the singular value decomposition of a matrix Z of order N × m when the rank (Z) = q ≤ 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. © 1997 Academic Press.