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

I use a parametric, bijective transformation to generate heavy tail versions Y of an arbitrary random variable (RV) X ~ F_X, by similar concepts as in Goerg (2011) for skewed RVs. The tail behavior of the so-called heavy tail Lambert W x F_X 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 used to remove skewness and heavy tails from data and then apply standard methods and models to this so obtained "nice" (Gaussianized) data. The optimal inverse transformation can be estimated by maximum likelihood. This transformation based approach to heavy tails also yields analytical, concise and simple expressions for the cumulative distribution (cdf) G_Y(y) and probability density function (pdf) g_Y(y). As a special case, I present explicit expressions for Tukey's h pdf and cdf - to the authors knowledge for the first time in the literature. Applications to a simulated Cauchy sample, S&P 500 log-returns, as well as solar flares data demonstrate the usefulness of the introduced methodology. The R package "LambertW" (cran.r-project.org/web/packages/LambertW) contains a wide range of methods presented here and is publicly available at CRAN