Tailoring Density Functions via Orthogonal Polynomials

Abstract

This paper re-examines the issue of how to tailor distributions to embody evidence of moments and dependence structure deviating from those of a given parent distribution. It is well known that the function that achieves the transformation from a given parent to a target distribution can be expressed as a linear combination of a set of orthogonal polynomials associated to parent distribution. We show that the coefficients of such linear combination are simple algebraic functions of the difference between the moments of the parent and target distributions. These results facilitate reshaping a considerably broader class of distributions than well-known approaches based on Hermite polynomials, which can be used only to reshape the normal distribution and allow to do so only to a limited extent. We provide applications to modeling distributions of financial asset returns, which are known to exhibit considerable skewness and fat tails.