What Determines EP Curve Shape?

Abstract

Propose use kurtosis divided by skewness squared as shape factor, and use the global or conditional minimum/maximum of this shape factor for selecting and differentiating distribution families. Semi-empirical formulas for that lower/upper bound are calculated for various distribution families, with the aid of Computer Algebra System, for fitting hard to match distributions. Previous studies show high CV distribution is hard to fit and simulate, this study extends that conclusion to cases with low CV but still hard to match EP curves, characterized by having shape factors close to 1. The maximal likelihood approach of distribution fit can tell us which distribution family is better suited for an empirical distribution, but the shape factor range information can tell us why a distribution cannot fit well, or is not suitable at all. So the shape factor, in a sense, determines the EP curve shape.

Shape Factor Asymptotic Analysis II

“…In signal processing, there is another definition of shape factor as the root mean square divided by the mean of the absolute value, https://www.mathworks.com/help/predmaint/ug/signal-features.html. This is a variant of CV, or the square root of the SF4[2] in Wang (2019a). We studied CV in Wang (2018a) and further research lead to our higher order shape factor definition that are more intrinsic to the shape of the distribution PDF, Wang (2018b and2019a).…”

“…This is a variant of CV, or the square root of the SF4[2] in Wang (2019a). We studied CV in Wang (2018a) and further research lead to our higher order shape factor definition that are more intrinsic to the shape of the distribution PDF, Wang (2018b and2019a). The CV, Gini index, and normalized skewness, defined as the sign-keeping square root of the reciprocal of the shape factor, are indeed related to each other from our empirical study, which showed that CV is almost identical to Gini index, and are the base part or lower envelope of the normalized skewness; these are topics for a different research thesis.…”

“…This is the continuation of the study in Wang (2020) for shape factor, which is defined in Wang (2019a) with a plan of addressing probability distribution fitting validation and explanation problem by examining shape factor global and conditional minimum or maximum formulas or curves, to other distribution families. The problems include, to name a few, why it is not able to fit a data distribution by some distribution families, and why different distribution families can fit a data distribution almost as good as each other while their PDF or even distribution domains are very different.…”

“…We will refer freely on contents of Wang (2019a and2020) rather than repeat it here, to avoid self-copying. Distributions definitions and parameter assumptions will refer to Marichev and Trott (2013).…”