Natural Exponential Families and Generalized Hypergeometric Measures
Let be a positive Borel measure on ℝ and (, . . . , ; , . . . , ; ) be a generalized hypergeometric series. We define a := (, . . . , ; , . . . , ), as a series of convolution powers of the measure , and we investigate probability distributions which are expressible as generalized hypergeometric measures. We show that the family of distributions defined by Kemp (, Ser. A, 30, (1968), 401 - 410) are examples of generalized hypergeometric measures in which is a Dirac measure on ℝ. For the case in which is a Dirac measure on ℝ, we relate the generalized hypergeometric measures to the diagonal natural exponential families classified by Bar-Lev, . (. 7 (1994), 883-929). For we show that certain generalized hypergeometric measures can be expressed as the convolution of a sequence of independent multidimensional Bernoulli trials. For = + 1, we show that the generalized hypergeometric measures µ are mixture measures with the Dufresne and Poisson-stopped-sum probability distributions as their mixing measures