Orthogonal polynomials for tailoring density functions to excess kurtosis, asymmetry, and dependence

Abstract

This article deals with the problem of tailoring distributions to embody evidence of moments and dependence structure deviating from those of a given parent law. First, we show that finite-moment distributions can be reshaped, to allow for extra kurtosis, asymmetry, and dependence by using orthogonal polynomials. Then, we derive a set of orthogonal polynomials for adjusting any symmetric density to given requirements in terms of moments. Conditions for positiveness of the resulting polynomially modified distribution are further established. This provides a broader approach to reshaping parent distributions by means of polynomial adjustments than that currently found in the literature.

Keywords:

• Asymmetry
• Between-squares dependence
• Kurtosis
• Orthogonal Polynomials vs. Moments

MATHEMATICS SUBJECT CLASSIFICATION:

• 33C45
• 62E10