Details

Autor(en) / Beteiligte

• Steerneman, A.G.M.
• van Perlo-ten Kleij, Frederieke

Titel

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

Ist Teil von

• Statistics & probability letters, 2008-02, Vol.78 (3), p.249-253

Ort / Verlag

Amsterdam: Elsevier B.V

Erscheinungsjahr

2008

Link zum Volltext

Quelle

Elsevier ScienceDirect Journals Complete

Beschreibungen/Notizen

• If X ∼ N n × k ( M , I n ⊗ Σ ) , then S = X ′ X has the noncentral Wishart distribution W k ′ ( n , Σ ; Λ ) , where Λ = M ′ M . Here Σ is allowed to be singular. It is well known that if Λ = 0 , then S has a (central) Wishart distribution and S is positive definite with probability 1 if and only if n ⩾ k and Σ 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 ⩾ k and Σ + Λ is positive definite. This is a consequence of the main result that rank X = min ( n , rank ( Σ + Λ ) ) with probability 1.