Orthogonal polynomials in the Cumulative Ord family and its application to variance bounds

TL;DR

This paper explores the properties of the Cumulative Ord family of distributions, focusing on orthogonal polynomials, and develops variance bounds for functions of these distributions that unify and improve existing results.

Contribution

It provides a complete classification of the Cumulative Ord family via orthogonal polynomials and derives new variance bounds based on forward differences, enhancing previous methods.

Findings

Classification of the Ord family via orthogonality of Rodrigues polynomials

Relationships between Fourier coefficients of functions and their forward differences

New variance bounds that unify and improve existing results

Abstract

This article presents and reviews several basic properties of the Cumulative Ord family of distributions; this family contains all the commonly used discrete distributions. A complete classification of the Ord family of probability mass functions is related to the orthogonality of the corresponding Rodrigues polynomials. Also, for any random variable $X$ of this family and for any suitable function $g$ in $L^2(\mathbb{R},X)$, the article provides useful relationships between the Fourier coefficients of $g$ (with respect to the orthonormal polynomial system associated to $X$) and the Fourier coefficients of the forward difference of $g$ (with respect to another system of polynomials, orthonormal with respect to another distribution of the system). Finally, using these properties, a class of bounds for the variance of $g(X)$ is obtained, in terms of the forward differences of $g$. These bounds unify and improve several existing results.