Haight's distributions as a natural exponential family
In an index to the distributions of mathematical statistics, \textit{F. A. Haight} [J. Res. Nat. Bureau of Standards 65B(1), 23-60 (1961)] considers, without giving any references, the following distribution: \[ \alpha^{-1}\exp (-xe^{\alpha}\alpha^{- 1})\sum^{\infty}_{n=0}(n+1)^{n-1}(n!)^{-2}x^ n\mathbf{1}_{(0,\infty)}(x)dx\quad for\quad 0<\alpha <1. \] We show that these distributions belong to a large class of natural exponential families for which we compute the variance functions. These computations are then used to prove the infinite divisibility of Haight's distribution.
scientific article; zbMATH DE number 4030698