Unifying the Named Natural Exponential Families and Their Relatives

Five of the six univariate natural exponential families (NEFs) with quadratic variance functions (QVFs), meaning that their variances are at most quadratic functions of their means, are the Normal, Poisson, Gamma, Binomial, and Negative Binomial distributions. The sixth is the NEF-CHS, the NEF generated from convolved Hyperbolic Secant distributions. These six NEF-QVFs and their relatives are unified in this article and in the main diagram via arrows that connect NEFs with many other named distributions. Relatives include all of Pearson's families of conjugate distributions (e.g., Inverted Gamma, Beta, F, and Skewed-t), conjugate mixtures (including two Polya urn schemes), and conditional distributions (including Hypergeometrics and Negative Hypergeometrics). Limit laws that also relate these distributions are indicated by solid arrows in Figure 1.