The Lambert Way to Gaussianize Heavy-Tailed Data with the Inverse of Tukey’shTransformation as a Special Case

Abstract

I present a parametric, bijective transformation to generate heavy tail versions of arbitrary random variables. The tail behavior of thisheavy tail Lambert  W × FXrandom variable depends on a tail parameterδ≥0: forδ=0,Y≡X, forδ>0 Yhas heavier tails thanX. ForXbeing Gaussian it reduces to Tukey’shdistribution. The Lambert W function provides an explicit inverse transformation, which can thus remove heavy tails from observed data. It also provides closed-form expressions for the cumulative distribution (cdf) and probability density function (pdf). As a special case, these yield analytic expression for Tukey’shpdf and cdf. Parameters can be estimated by maximum likelihood and applications to S&P 500 log-returns demonstrate the usefulness of the presented methodology. The R packageLambertWimplements most of the introduced methodology and is publicly available onCRAN.