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.

Keywords: Skewness, kurtosis, TVaR, shape factor, reinsurance, computer algebra system, Beta distribution, Kumaraswamy distribution, asymptotic expansion, GB2 distribution, numerical optimization, generalized hyperbolic distribution
