Paper detail

The Lambert Way to Gaussianize heavy tailed data with the inverse of Tukey's h as a special case

I present a parametric, bijective transformation to generate heavy tail versions Y of arbitrary RVs X ~ F. The tail behavior of the so-called 'heavy tail Lambert W x F' RV Y depends on a tail parameter delta >= 0: for delta = 0, Y = X, for delta > 0 Y has heavier tails than X. For X being Gaussian, this meta-family of heavy-tailed distributions reduces to Tukey's h distribution. Lambert's W function provides an explicit inverse transformation, which can be estimated by maximum likelihood. This inverse can remove heavy tails from data, and also provide analytical expressions for the cumulative distribution (cdf) and probability density function (pdf). As a special case, these yield explicit formulas for Tukey's h pdf and cdf - to the author's knowledge for the first time in the literature. Simulations and applications to S&P 500 log-returns and solar flares data demonstrate the usefulness of the introduced methodology. The R package "LambertW" (cran.r-project.org/web/packages/LambertW) implementing the presented methodology is publicly available at CRAN.