Wishart and Pseudo-Wishart Distributions and Some Applications to Shape Theory,

Suppose thatX~N-m([mu], [Sigma], [Theta]). An expression for the density function is given when[Sigma][greater-or-equal, slanted]0 and/or[Theta]:[greater-or-equal, slanted]0. An extension of Uhlig's result (Uhlig [17]) is expanded for the singular value decomposition of a matrixZof orderN-mwhen the rank (Z)=q[less-than-or-equals, slant]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.