One-dimensional exponential families with constant Hessian scalar curvature

Abstract

We give a complete classification of 1-dimensional exponential families $\mathcal{E}$ defined over a finite space $\Omega =\{x_0,...,x_n\}$ whose Hessian scalar curvature is constant. We observe an interesting phenomenon: if $\mathcal{E}$ has constant Hessian scalar curvature, say $\lambda$, then $\lambda =\frac{2}{k}$ for some positive integer $k\le m$. We also discuss the central role played by the binomial distribution in this classification.