Archive for David Hinkley

ratio of Gaussians

Posted in Books, Statistics, University life with tags , , , , , , , , on April 12, 2021 by xi'an

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.