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.

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


    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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