One-dimensional exponential families with constant Hessian scalar curvature

Abstract

We give a complete classification of 1-dimensional exponential families E defined over a finite space Ω = {x 0 , ..., x m } whose Hessian scalar curvature is constant. We observe an interesting phenomenon: if E has constant Hessian scalar curvature, say λ, then λ = 2 k for some positive integer k ≤ m. We also discuss the central role played by the binomial distribution in this classification.