Theoretical considerations when simulating data from the g‐and‐h family of distributions

Abstract

The g‐and‐h family of distributions is a computationally efficient, flexible option to model and simulate non‐normal data. In spite of its popularity, there are several theoretical aspects of these distributions that need special consideration when they are used. In this paper some of these aspects are explored. In particular, through mathematical analysis it is shown that a popular multivariate generalization of the g‐and‐h distribution may result in marginal distributions which are no longer g‐and‐h distributed, that more than one set of (g,h) parameters can correspond to the same values of population skewness and excess kurtosis, and that multivariate generalizations of g‐and‐h distributions available in the literature are special cases of Gaussian copula distributions. A small‐scale simulation is also used to demonstrate how simulation conclusions can change when different (g,h) parameters are used to simulate data, even if they imply the same population values of skewness and excess kurtosis.

Inferences About a Robust Heteroscedastic Measure of Effect Size When There Are Two Covariates

“…The choice (g, h) = (0, 0) corresponds to a standard normal distribution, (g, h) = (1, 0) corresponds to a lognormal distribution shifted to have a median of zero, while (g, h) = (0, 0.2) is a symmetric distribution about zero having kurtosis approximately equal to 24.8. For recent results on the properties of g-and-h distributions, see Astivia and Edward (2022).…”