Closed-form cdf and pdf of Tukey's h-distribution, the heavy-tail Lambert W approach, and how to bijectively "Gaussianize" heavy-tailed data

Recently Goerg (2010) introduced Lambert W - F random variables (RVs), a new family of generalized skewed distributions. Here I will adapt this appealing framework to generate heavy (heavier) tailed versions of arbitrary distributions. As in the skewed case a non-linear, parametric transformation of an input RV X with arbitrary cumulative distribution function (cdf) FX(x) yields a heavy-tailed version Y . The tail behavior depends on a tail parameter gamma > = 0; for gamma = 0, Y = X, for gamma > 0 Y has heavier tails than X. It turns out that heavy-tail Lambert W - Gaussian RVs equals heavy-tailed Tukey h RVs (the g - h family with g = 0), with the major advantage that the Lambert W framework gives closed-form solution of the inverse transformation of the h-transformation, and thus a analytical expressions for the cdf and pdf for Tukey's h distribution - to the author's knowledge the ?rst time in the literature. Furthermore, the Lambert W approach allows practicioners to "Gaussianize" their heavy-tailed data and apply common methods and models on the latent Gaussian RV. The optimal parameters to do the backtransformation can be estimated by maximum likelihood (ML). Contrary to the skewed case, the transformation is bijective: each observed data point is uniquely linked to its hidden (and normally tailed) input. These extensions will soon be added to the LambertW R package, originally implemented for the skew Lambert W case.