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

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 $g\left(X\right)$ is obtained, in terms of the forward differences of g. These bounds unify and improve several existing results.

KEYWORDS:

• Cumulative Ord family
• Fourier coefficients
• orthogonal polynomials
• Rodrigues-type formula
• variance bounds

AMS SUBJECT CLASSIFICATION:

• Primary: 60E05
• 62E99
• 05E35
• 42A61
• Secondary:60E15

Acknowledgments

The authors acknowledge the editorial team who handled the paper for providing suggestions that resulted in improving the presentation of the results.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research has been co-financed by the European Union (European Social Fund – ESF) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF) – Research Funding Program: ARISTEIA, Grant No. 4357. Also, this work is partially supported by the University of Athens Research Grant 70/4/5637 and by internal funds, Department of Biostatistics, SUNY Buffalo. This work was also partially supported by the Natural Sciences and Engineering Research Council of Canada through an Individual Discovery Grant to the second author.