The rank of a normally distributed matrix and positive definiteness of a noncentral Wishart distributed matrix

A.G.M. Steerneman and Frederieke van Perlo-ten Kleij

Statistics & Probability Letters, 2008, vol. 78, issue 3, 249-253

Abstract: If , then S=X'X has the noncentral Wishart distribution , where [Lambda]=M'M. Here [Sigma] is allowed to be singular. It is well known that if [Lambda]=0, then S has a (central) Wishart distribution and S is positive definite with probability 1 if and only if n[greater-or-equal, slanted]k and [Sigma] is positive definite. We show that if S has a noncentral Wishart distribution, then S is positive definite with probability 1 if and only if n[greater-or-equal, slanted]k and [Sigma]+[Lambda] is positive definite. This is a consequence of the main result that with probability 1.

Keywords: Multivariate; normal; distribution; Singular; covariance; matrix; Rank; Noncentral; Wishart; Positive; definite (search for similar items in EconPapers)
Date: 2008
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