A guy I know wrote a paper titled The Lambert Way to Gaussianize Heavy-Tailed Data with the Inverse of Tukey’s h Transformation as a Special Case and has an associated R package LambertW. The TL;DR is that the paper has a parameterization to remove skewness and left/right kurtosis from random variables that reminds me a lot of non-centered parametrizations.
The package has a good example of this
# Univariate example library(LambertW) set.seed(20) y1 <- rcauchy(n = 100) plot(density(y1)) out <- Gaussianize(y1, return.tau.mat = TRUE) plot(density(out$input) # huh! x1 <- get_input(y1, c(out$tau.mat[, 1])) # same as out$input test_normality(out$input) # Gaussianized a Cauchy!
Wanted to throw it out there in case anyone though it would be a useful tool. On the backend we would need to add boosts LambertW implementation and just coding over the stuff in the R package to stan math