McCullagh's parametrization of the Cauchy distributions

In probability theory, the "standard" Cauchy distribution is the probability distribution whose probability density function is


 * $$f(x) = {1 \over \pi (1 + x^2)}$$

for x real. This has median 0, and first and third quartiles respectively &minus;1 and +1. Generally, a Cauchy distribution is any probability distribution belonging to the same location-scale family as this one. Thus, if X has a standard Cauchy distribution and &mu; is any real number and &sigma; > 0, then Y = &mu; + &sigma;X has a Cauchy distribution whose median is &mu; and whose first and third quartiles are respectively &mu; &minus; &sigma; and &mu; + &sigma;.

McCullagh's parametrization, introduced by Peter McCullagh, professor of statistics at the University of Chicago, assigns this last distribution to the complex number &theta; = &mu; + i&sigma;, where i is the imaginary unit. To the question "Why introduce complex numbers when only real-valued random variables are involved?", McCullagh wrote:

To this question I can give no better answer than to present the curious result that


 * $Y^* = {aY + b \over cY + d} \sim C\left({a\theta + b \over c\theta + d}\right)$

for all real numbers a, b, c and d. ...the induced transformation on the parameter space has the same fractional linear form as the transformation on the sample space only if the parameter space is taken to be the complex plane.

In other words, if the random variable Y has a Cauchy distribution with complex parameter &theta;, then the random variable Y* defined above has a Cauchy distribution with parameter (a&theta; + b)/(c&theta; + d).

He also wrote, "The distribution of the first exit point from the upper half-plane of a Brownian particle starting at &theta; is the Cauchy density on the real line with parameter &theta;."

Reference

 * Peter McCullagh, "Conditional inference and Cauchy models", Biometrika, volume 79 (1992), pages 247 - 259. PDF from McCullagh's homepage.