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

Suppose thatX∽NN×m(μ, Σ, Θ). An expression for the density function is given whenΣ⩾0 and/orΘ:⩾0. An extension of Uhligʹs result (Uhlig [17]) is expanded for the singular value decomposition of a matrixZof orderN×mwhen 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.