ratio of Gaussians

Following (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.

4 Responses to “ratio of Gaussians”

1. Gerard Letac Says:

If $w=a+ib$ with $b\ge ;0$ the Cauchy distribution $c_w(dx)=bdx/(\pi(x-a)^2+b^2))$ has the following property: for $h(x)=(\alpha x+\beta)/(\gamma x+\delta)$ with the determinant $\alpha\delta-\beta \gamma\ge 0$ then $c({h(w)}\sim h(X)$ if $X\sim c_{w}.$ This result can be found in a magnificent paper completely ignored by the statisticians ‘Which functions preserve Cauchy Laws’ Proceedings of the AMS 1977.

• Gerard Letac Says:

I mean that if $Z_1,Z_2$ are iid N(0,1) then the distribution of
$(\alpha Z_1+\beta Z_2+ c)/(\gamma Z_1+\delta Z_2+d)$ is Cauchy with complex parameter $(\alpha \iota+\beta/)(\gamma \iota+\delta)$ )if $c=d=0.$ No elegance indeed if not.

• A complex comment!!!

2. 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).

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