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.

This entry was posted on April 12, 2021 at 12:21 am and is filed under Books, Statistics, University life with tags Biometrika, Cauchy distribution, cross validated, David Hinkley, Edgar Fieller, George Marsaglia, JASA, mixtures of distributions, ratio of random variates. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

4 Responses to “ratio of Gaussians”

Gerard Letac Says:
April 12, 2021 at 12:54 pm

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.

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- Gerard Letac Says:
  April 12, 2021 at 1:13 pm
  
  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.
  
  Reply
  - xi'an Says:
    April 14, 2021 at 1:58 pm
    
    A complex comment!!!
FJRubio Says:
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).

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