# Lambert W random variables - a new family of generalized skewed distributions with applications to risk estimation

## TL;DR

This paper introduces Lambert W random variables, a new flexible family of skewed distributions derived from a nonlinear transformation, with applications in risk estimation, and provides estimation methods and practical tools for data unskewing.

## Contribution

It presents a novel class of distributions based on Lambert W transformations, including estimation procedures and real-world applications.

## Key findings

- Distribution and density functions are variants of input distributions.
- Estimation of skewness parameter $\gamma$ does not impair other parameter estimates.
- Applications demonstrate usefulness for modeling slightly skewed data.

## Abstract

Originating from a system theory and an input/output point of view, I introduce a new class of generalized distributions. A parametric nonlinear transformation converts a random variable $X$ into a so-called Lambert $W$ random variable $Y$, which allows a very flexible approach to model skewed data. Its shape depends on the shape of $X$ and a skewness parameter $\gamma$. In particular, for symmetric $X$ and nonzero $\gamma$ the output $Y$ is skewed. Its distribution and density function are particular variants of their input counterparts. Maximum likelihood and method of moments estimators are presented, and simulations show that in the symmetric case additional estimation of $\gamma$ does not affect the quality of other parameter estimates. Applications in finance and biomedicine show the relevance of this class of distributions, which is particularly useful for slightly skewed data. A practical by-result of the Lambert $W$ framework: data can be "unskewed." The $R$ package http://cran.r-project.org/web/packages/LambertWLambertW developed by the author is publicly available (http://cran.r-project.orgCRAN).