The mean radius of curvature of an exponential family

Abstract

Let F=F([mu]) be the natural exponential family (NEF) on the Euclidean space generated by the measure [mu] and let k[mu]=log L[mu] be the log-Laplace transform of [mu]. In this paper, a notion of mean radius of curvature function for the NEF F is introduced using the mean radius of curvature function of the convex epigraph of the function k[mu]. An algebraic property for the variance function of F is deduced and some characteristic properties for the family F related to the mean radius of curvature function are discussed. The results are illustrated by some examples.Exponential family Variance function Support function