The diagnonal multivariate natural exponential families and their classification

A natural exponential family (NEF)F in ℝⁿ, n>1, is said to be diagonal if there exist n functions, a₁,..., a_n, on some intervals of ℝ, such that the covariance matrix V_F(m) of F has diagonal (a₁(m₁),..., a_n(m_n)), for all m=(m₁,..., m_n) in the mean domain of F. The family F is also said to be irreducible if it is not the product of two independent NEFs in ℝ^k and ℝ^n-k, for some k=1,..., n-1. This paper shows that there are only six types of irreducible diagonal NEFs in ℝⁿ, that we call normal, Poisson, multinomial, negative multinomial, gamma, and hybrid. These types, with the exception of the latter two, correspond to distributions well established in the literature. This study is motivated by the following question: If F is an NEF in ℝⁿ, under what conditions is its projection p(F) in ℝ^k, under p(x₁,..., x_n):=(x₁,..., x_k), k=1,..., n-1, still an NEF in ℝ^k? The answer turns out to be rather predictable. It is the case if, and only if, the principal k×k submatrix of V_F(m₁,..., m_n) does not depend on (m_k+1,..., m_n).