Some Utilizations of Lambert W Function in Distribution Theory

We introduce a new class of positive infinitely divisible probability laws calling them ? 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 ? laws are bounded resembling pdf of a Levy stable distribution. The exponential dispersion model constructed starting from an ? distribution admits the inverse Gaussian approximation. The natural exponential family constructed starting from an ? distribution constitutes the reciprocal of the natural exponential family generated by a spectrally negative stable law with =1. We derive new results on ? 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 ? 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.