ABSTRACT
In other words, dm*(s) is an invariant measure on the space of S under the action a of the group of transformations defined by s? CsC', Now (6.30) follows from the uniqueness of invariant measures on homogeneous spaces (see Nachbin, 1965; or Eaton, 1972). From (6.24) and (6.30) the probability density
function Wp(n, S) of a Wishart random variable S with m degrees of freedom and parameter S is given by (with respect to the Lebesgue measure ds)
where K is the normalizing constant independent of S. To specify the probability density function we need to evaluate the constant K. Since K is independent of S, we can in particular take S=I for the evaluation of K. Since K is a function of n and p, we shall denote it by Cn,p. Let us partition S=(Sij) as
with S(11) a (p-1)×(p-1)) submatrix of S, and let
From (6.32)
as
Since Wp(n, I) is a probability density function with the constant K=Cn,p, we obtain
From (6.33) and (6.34) we get
But Cn,1 is given by
that implies
From (6.35) and (6.36) we get
The derivation of the Wishart distribution, which is very fundamental in multivariate analysis, was a major breakthrough for the development of multivariate analysis. Several derivations of the Wishart distribution are available in the literature. The derivation given here involves a property of invariant measure and is quite short and simple in nature.
