The alchemy of probability distributions: beyond Gram-Charlier expansions, and a skew-kurtotic-normal distribution from a rank transmutation map

TL;DR

This paper introduces a novel transmutation map technique to generate rich, parametric skewed and kurtotic distributions without relying on asymptotic expansions, enhancing tractability for financial and statistical applications.

Contribution

It presents a new method for creating skewed and kurtotic distributions via composition of distribution functions, avoiding Gram-Charlier expansion limitations.

Findings

Generated various skewed distributions like skew-normal and skew-kurtotic-normal.

Avoided pathologies associated with asymptotic expansion methods.

Provided flexible parametric families for financial modeling and statistics.

Abstract

Motivated by the need for parametric families of rich and yet tractable distributions in financial mathematics, both in pricing and risk management settings, but also considering wider statistical applications, we investigate a novel technique for introducing skewness or kurtosis into a symmetric or other distribution. We use a "transmutation" map, which is the functional composition of the cumulative distribution function of one distribution with the inverse cumulative distribution (quantile) function of another. In contrast to the Gram-Charlier approach, this is done without resorting to an asymptotic expansion, and so avoids the pathologies that are often associated with it. Examples of parametric distributions that we can generate in this way include the skew-uniform, skew-exponential, skew-normal, and skew-kurtotic-normal.