Moment-based Hermite model for asymptotically small non-Gaussianity

Abstract

The third degree Moment-based Hermite model, which expresses a random variable as a cubic transformation of a standard normal variable, offers versatility in engineering applications. While its probability density function is not directly tractable, it is more complex to compute than the Gram-Charlier series, which, despite its simplicity, suffers from limitations such as positivity and unimodality issues, restricting its range of applicability. This paper presents two asymptotic analyses of the cubic Moment-based Hermite model for slight non-Gaussianity (i.e. small skewness and excess coefficients, “small” being understood in the sense of perturbation methods), showing that it asymptotically converges to the fourth cumulant Gram-Charlier model, while offering a slightly broader domain of applicability with minimal additional computational cost. Additionally, the paper derives, mathematically, a non empirical expression for the monotone limit of the original cubic translation model, and validates the theoretical findings through numerical simulations.