Real natural exponential families and generalized orthogonality

Abstract

In this article, we use the notion of generalized orthogonality for a sequence of polynomials introduced by Bryc, Fakhfakh, and Mlotkowski (2019) to extend the characterizations of the Feinsilver, Meixner, and Shanbhag based on orthogonal polynomials. These new versions subsume the real natural exponential families (NEFs) having polynomial variance function in the mean of arbitrary degree. We also relate generalized orthogonality to Sheffer systems. We show that the generalized orthogonality of Sheffer systems occurs if and only if the associated classical additive convolution semigroup of probability measures generates NEFs with polynomial variance function. In addition, we use the raising and lowering operators for quasi-monomial polynomials associated with NEFs to give a characterization of NEFs with polynomial variance function of arbitrary degree.