LambertW-package: R package for Lambert W \times F distributions

LambertW-packageR Documentation

R package for Lambert W \times F distributions


This package is based on notation, definitions, and results of Goerg (2011, 2015, 2016). I will not include these references in the description of each single function.

Lambert W \times F distributions are a general framework to model and transform skewed, heavy-tailed data. Lambert W \times F random variables (RV) are based on an input/ouput system with input RV X \sim F_X(x \mid \boldsymbol β) and output Y, which is a non-linearly transformed version of X – with similar properties to X, but slightly skewed and/or heavy-tailed. Then Y has a 'Lambert W \times F_X' distribution - see References.

get_distnames lists all implemented Lambert W \times F distributions in this package. If you want to generate a skewed/heavy-tailed version of a distribution that is not implemented, you can use the do-it-yourself modular toolkit (create_LambertW_input and create_LambertW_output). It allows users to quickly implement their own Lambert W x 'MyFavoriteDistribution' and use it in their analysis right away.

This package contains several functions to analyze skewed and heavy-tailed data: simulate random samples (rLambertW), evaluate pdf and cdf (dLambertW and pLambertW), estimate parameters (IGMM and MLE_LambertW), compute quantiles (qLambertW), and plot/print results nicely (plot.LambertW_fit, print.LambertW_fit, summary.LambertW_fit).

Probably the most useful function is Gaussianize, which works similarly to scale, but makes your data Gaussian (not just centers and scales it, but also makes it symmetric and removes excess kurtosis).

If you use this package in your work please cite it (citation("LambertW")). You can also send me an implementation of your 'Lambert W \times YourFavoriteDistribution' to add to the LambertW package (and I will reference your work introducing your 'Lambert W \times YourFavoriteDistribution' here.)

Feel free to contact me for comments, suggestions, code improvements, implementation of new input distributions, bug reports, etc.


Author and maintainer: Georg M. Goerg (im (at)


Goerg, G.M. (2011). “Lambert W Random Variables - A New Family of Generalized Skewed Distributions with Applications to Risk Estimation”. Annals of Applied Statistics, 5 (3), 2197-2230. (

Goerg, G.M. (2015). “The Lambert Way to Gaussianize heavy-tailed data with the inverse of Tukey's h transformation as a special case”. The Scientific World Journal: Probability and Statistics with Applications in Finance and Economics. Available at

Goerg, G.M. (2016). “Rebuttal of the “Letter to the Editor of Annals of Applied Statistics” on Lambert W x F distributions and the IGMM algorithm”. Available on arxiv.


## Not run: 
# Replicate parts of the analysis in Goerg (2011)
y <- AA[AA$sex=="f", "bmi"]

fit.gmm <- IGMM(y, type = "s")
summary(fit.gmm)  # gamma is significant and positive

# Compare empirical to theoretical moments (given parameter estimates)
moments.theory <-
 mLambertW(theta = list(beta = fit.gmm$tau[c("mu_x", "sigma_x")],
                        gamma = fit.gmm$tau["gamma"]),
           distname = "normal")
TAB <- rbind(unlist(moments.theory),
             c(mean(y), sd(y), skewness(y), kurtosis(y)))
rownames(TAB) <- c("Theoretical (IGMM)", "Empirical")

x <- get_input(y, fit.gmm$tau)
test_normality(x) # input is normal -> fit a Lambert W x Gaussian by MLE <- MLE_LambertW(y, type = "s", distname = "normal", hessian = TRUE)

## End(Not run)

LambertW documentation built on Sept. 22, 2022, 5:07 p.m.