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September 2011 Lambert W random variables—a new family of generalized skewed distributions with applications to risk estimation
Georg M. Goerg
Ann. Appl. Stat. 5(3): 2197-2230 (September 2011). DOI: 10.1214/11-AOAS457


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).


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Georg M. Goerg. "Lambert W random variables—a new family of generalized skewed distributions with applications to risk estimation." Ann. Appl. Stat. 5 (3) 2197 - 2230, September 2011.


Published: September 2011
First available in Project Euclid: 13 October 2011

zbMATH: 1228.62016
MathSciNet: MR2884937
Digital Object Identifier: 10.1214/11-AOAS457

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

Rights: Copyright © 2011 Institute of Mathematical Statistics


Vol.5 • No. 3 • September 2011
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