Distributional Characteristics: Just a Few More Moments

Abstract

The ability of statistical models to accurately characterize distributional characteristics such as skewness and kurtosis can impact the results of statistical analysis. This article compares the feasible skewness–kurtosis spaces for two generalizations of the lognormal, the inverse hyperbolic sine (IHS) and g-and-h probability density functions (pdf’s), each of which can accommodate a wide variety of distributional characteristics. For h ≥ 0, the boundary of the skewness–kurtosis spaces for g-and-h and IHS coincides with that of a generalized (three-parameter) lognormal (LN*) distribution. The increased skewness–kurtosis flexibility of the g-and-h distribution, for h < 0, relative to the IHS is obtained by introducing vertical asymptotes, compact support, and possibly U-shaped pdf’s. This increased coverage, however, may not be helpful if the data are unimodal. Empirical daily, weekly, and monthly stock return data are used to compare the descriptive ability of the IHS, g-and-h, and LN* distributions.