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

✍ Scribed by José A Dı́az-Garcı́a; Ramón Gutierrez Jáimez; Kanti V Mardia

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 321 KB
• Volume: 63
• Category: Article
• ISSN: 0047-259X
• DOI: 10.1006/jmva.1997.1689

✦ Synopsis

Suppose that XtN N_m (+, 7, 3). An expression for the density function is given when 7 0 andÂor 3 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

1. Introduction

Let XtN N_m (+, 7, I N ), where 7 is positive definite, 7>0 of order m_m and N m. Then, the random matrix V=X$X has a noncentral nonsingular Wishart distribution W m (N, 7, 0), with a matrix of noncentrality parameters defined by 0=7 &1 +$+ (see Muirhead [12, p.