Orthogonal polynomials in the cumulative Ord family and its application to variance bounds. Issue 2 (4th March 2018)

• Record Type: Journal Article
• Authors: Afendras, Georgios; Balakrishnan, Narayanaswamy; Papadatos, Nickos
• Is Part Of: Statistics. Volume 52:Issue 2(2018)
• Page Start: 364
• Page End: 392
• Publication Date: 2018-03-04
• Subjects: Cumulative Ord family -- Fourier coefficients -- orthogonal polynomials -- Rodrigues-type formula -- variance bounds; Primary: 60E05 -- 62E99 -- 05E35 -- 42A61 -- Secondary:60E15; Mathematical statistics -- Periodicals; 519.505
• DOI: 10.1080/02331888.2017.1406940
• Languages: English
• ISSNs: 0233-1888

Abstract

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 inL 2 ( 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 ofg ( X ) is obtained, in terms of the forward differences of g . These bounds unify and improve several existing results.