Some Utilizations of Lambert W Function in Distribution Theory

Abstract

We introduce a new class of positive infinitely divisible probability laws calling them 𝔏γ distributions. Their cumulant-generating functions (cgf) are expressed in terms of the principal branch of the Lambert W function. The probability density functions (pdfs) of 𝔏γ laws are bounded resembling pdf of a Lévy stable distribution. The exponential dispersion model constructed starting from an 𝔏γ distribution admits the inverse Gaussian approximation. The natural exponential family constructed starting from an 𝔏γ distribution constitutes the reciprocal of the natural exponential family generated by a spectrally negative stable law with α = 1. We derive new results on 𝔏γ laws and the related exponential dispersion models, including their convolution and scaling closure properties. We generate another exponential dispersion model starting from an exponentially compounded 𝔏γ law. This distribution emerges in the Poisson mixture representation of a generalized Poisson law. We extend the Poisson approximation for the scaled Neyman type A exponential dispersion model. We derive saddlepoint-type approximations for some of these exponential dispersion models. The role of the Lambert W function is emphasized.