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

Abstract EN

I present a parametric, bijective transformation to generate heavy tail versions of arbitrary random variables.

The tail behavior of this heavy tail Lambert W × F X random variable depends on a tail parameter δ ≥ 0 : for δ = 0 , Y ≡ X , for δ > 0 Y has heavier tails than X .

For X being Gaussian it reduces to Tukey’s h distribution.

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’s h pdf 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 package LambertW implements most of the introduced methodology and is publiclyavailable on CRAN.