## ratio of Gaussians

**F**ollowing (as usual) an X validated question, I came across two papers of George Marsaglia on the ratio of two arbitrary (i.e. unnormalised and possibly correlated) Normal variates. One was a 1965 JASA paper,

where the density of the ratio X/Y is exhibited, based on the fact that this random variable can always be represented as (a+ε)/(b+ξ) where ε,ξ are iid N(0,1) and a,b are constant. Surprisingly (?), this representation was challenged in a 1969 paper by David Hinkley (corrected in 1970).

And less surprisingly the ratio distribution behaves almost like a Cauchy, since its density is

meaning it is a two-component mixture of a Cauchy distribution, with weight exp(-a²/2-b²/2), and of an altogether more complex distribution ƒ². This is remarked by Marsaglia in the second 2006 paper, although the description of the second component remains vague, besides a possible bimodality. (It could have a mean, actually.) The density ƒ² however resembles (at least graphically) the generalised Normal inverse density I played with, eons ago.

April 12, 2021 at 12:54 pm

If with the Cauchy distribution has the following property: for with the determinant then if This result can be found in a magnificent paper completely ignored by the statisticians ‘Which functions preserve Cauchy Laws’ Proceedings of the AMS 1977.

April 12, 2021 at 1:13 pm

I mean that if are iid N(0,1) then the distribution of

is Cauchy with complex parameter )if No elegance indeed if not.

April 14, 2021 at 1:58 pm

A complex comment!!!

April 12, 2021 at 11:52 am

Some time ago, I created an RPubs with the implementation of the density of the ratio of two independent normal variables:

https://rpubs.com/FJRubio/RatioNormals

to illustrate the shapes of this density, and the cases where (and it what sense) it could be approximated with a normal distribution (https://link.springer.com/article/10.1007/s00362-012-0429-2).