Modeling probability density functions of non-negative random variables using novel series expansions based on mellin kind statistics

Abstract

Mellin kind (MK) statistics is the framework which arises if the Fourier transform is replaced with the Mellin transform when computing the characteristic function from the probability density function. We may then proceed to retrieve logarithmic moments and cumulants, which have important applications in the analysis of heavy-tailed distribution models for non-negative random variables. In this paper we present a framework for series expansion methods based on MK statistics, which turn out to be strong analogies of classical Gram-Charlier and Edgeworth series. We introduce the MK Gram-Charlier series with arbitrary kernel, and evaluate the specific cases of the gamma, beta prime and log-normal kernel PDFs. We provide a derivation and expression of the MK Edgeworth series which has a few significant differences from its original reference. We conduct a broad numerical investigation as to the performance of the MK series expansions in approximating and estimating several target distributions, and in fitting real-world data. We focused on distributions which are relevant in radar imagery, but their general nature allows us to draw conclusions which apply to all non-negative random phenomena. We also applied the Bell polynomials to the classical series expansions. These are, to our knowledge, the first explicit expressions of the classical Gram-Charlier and Edgeworth series, which are more than a century old.