Annals of Applied Statistics

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

Georg M. Goerg

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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 γ. In particular, for symmetric X and nonzero γ 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 γ 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 LambertW developed by the author is publicly available (CRAN).

Article information

Source
Ann. Appl. Stat., Volume 5, Number 3 (2011), 2197-2230.

Dates
First available in Project Euclid: 13 October 2011

Permanent link to this document
https://projecteuclid.org/euclid.aoas/1318514301

Digital Object Identifier
doi:10.1214/11-AOAS457

Mathematical Reviews number (MathSciNet)
MR2884937

Zentralblatt MATH identifier
1228.62016

Keywords
Family of skewed distributions skewness transformation of random variables Lambert W latent variables stylized facts of asset returns value at risk GARCH

Citation

Goerg, Georg M. Lambert W random variables—a new family of generalized skewed distributions with applications to risk estimation. Ann. Appl. Stat. 5 (2011), no. 3, 2197--2230. doi:10.1214/11-AOAS457. https://projecteuclid.org/euclid.aoas/1318514301


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