## g-and-k [or -h] distributions

**D**ennis Prangle released last week an R package called gk and an associated arXived paper for running inference on the g-and-k and g-and-h quantile distributions. As should be clear from an earlier review on Karian’s and Dudewicz’s book quantile distributions, I am not particularly fond of those distributions which construction seems very artificial to me, as mostly based on the production of a closed-form quantile function. But I agree they provide a neat benchmark for ABC methods, if nothing else. However, as recently pointed out in our Wasserstein paper with Espen Bernton, Pierre Jacob and Mathieu Gerber, and explained in a post of Pierre’s on Statisfaction, the pdf can be easily constructed by numerical means, hence allows for an MCMC resolution, which is also a point made by Dennis in his paper. Using the closed-form derivation of the Normal form of the distribution [i.e., applied to Φ(x)] so that numerical derivation is not necessary.

July 18, 2017 at 10:37 am

Thanks for the reference Francisco – I’ll add this to the paper if I get asked to revise it. The sinh-arcsinh transformation (4) looks particularly appealing.

After writing this package I feel – similarly to Christian – that g&k style distributions aren’t of great practical use. The extra flexibility doesn’t seem enough to make up for the difficulty of fitting them. But I’d be happy to be proved wrong!

However I do like the strategy of creating distributions by transforming random variables. There are some interesting ML papers which do this by composing many transformations (e.g. https://arxiv.org/abs/1605.08803). It would be nice to use transformations from the quantile distributions literature here, particularly to allow light/heavy tails.

July 17, 2017 at 11:42 am

I have the impression that g-and-h, g-and-k, and Lambert-W-like distributions were brought back to life since people found that they are useful to produce nice examples using ABC tools.

An interesting, relatively balanced, catalogue of more tractable parametric flexible distributions can be found in the following discussion paper:

http://onlinelibrary.wiley.com/doi/10.1111/insr.12055/abstract

July 18, 2017 at 10:23 am

Thank you. Do you know of an open access version for this paper?

July 18, 2017 at 9:47 pm

There is this one:

http://statistics.open.ac.uk/802576CB00593013/(httpInfoFiles)/FE2EE03EDE61AB8E80257C700040D68E/$file/review.pdf

Although I recall there are some small differences with the journal version.

The sinh-arcsinh transformation is quite interesting, indeed, although the induced distribution cannot capture high levels of skewness (https://arxiv.org/abs/1307.6021).