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