# McCullagh's parametrization of the Cauchy distributions

﻿
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 −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 μ is any real number and σ > 0, then Y = μ + σX has a Cauchy distribution whose median is μ and whose first and third quartiles are respectively μ − σ and μ + σ.

McCullagh's parametrization, introduced by Peter McCullagh, professor of statistics at the University of Chicago uses the two parameters of the non-standardised distribution to form a single complex-valued parameter, specifically, the complex number θ = μ + iσ, where i is the imaginary unit. It also extends the usual range of scale parameter to include σ < 0.

Although the parameter is notionally expressed using a complex number, the density is still a density over the real line. In particular the density can be written using the real-valued parameters μ and σ, which can each take positive or negative values, as

$f(x) = {1 \over \pi \left \vert \sigma \right \vert (1 + \frac{(x-\mu)^2}{\sigma^2} ) }\,,$

where the distribution is regarded as degenerate if σ = 0. An alternative form for the density can be written using the complex parameter θ = μ + iσ as

$f(x) = {\left \vert \Im{\theta} \right \vert \over \pi \left \vert x-\theta \right \vert^2} \,,$

where $\Im{\theta} = \sigma \,.$

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 θ, then the random variable Y * defined above has a Cauchy distribution with parameter ( + b)/( + d).

McCullagh also wrote, "The distribution of the first exit point from the upper half-plane of a Brownian particle starting at θ is the Cauchy density on the real line with parameter θ." In addition, McCullagh shows that the complex-valued parameterisation allows a simple relationship to be made between the Cauchy and the "circular Cauchy distribution".

## References

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• McCullagh — may refer to: Colm McCullagh, Gaelic Football player for County Tyrone Crawford McCullagh (1868–1948), Unionist politician in Northern Ireland David McCullagh, Irish journalist and author Declan McCullagh, American journalist and columnist for… …   Wikipedia

• Cauchy distribution — Not to be confused with Lorenz curve. Cauchy–Lorentz Probability density function The purple curve is the standard Cauchy distribution Cumulative distribution function …   Wikipedia

• Peter McCullagh — is an Irish statistician, originally from Plumbridge, Northern Ireland. He attended Birmingham University, and completed his Ph.D. at Imperial College London. He is currently the John D. MacArthur Distinguished Service Professor in the Department …   Wikipedia

• List of statistics topics — Please add any Wikipedia articles related to statistics that are not already on this list.The Related changes link in the margin of this page (below search) leads to a list of the most recent changes to the articles listed below. To see the most… …   Wikipedia

• List of probability topics — This is a list of probability topics, by Wikipedia page. It overlaps with the (alphabetical) list of statistical topics. There are also the list of probabilists and list of statisticians.General aspects*Probability *Randomness, Pseudorandomness,… …   Wikipedia

• List of mathematics articles (M) — NOTOC M M estimator M group M matrix M separation M set M. C. Escher s legacy M. Riesz extension theorem M/M/1 model Maass wave form Mac Lane s planarity criterion Macaulay brackets Macbeath surface MacCormack method Macdonald polynomial Machin… …   Wikipedia

• Negative binomial distribution — Probability mass function The orange line represents the mean, which is equal to 10 in each of these plots; the green line shows the standard deviation. notation: parameters: r > 0 number of failures until the experiment is stopped (integer,… …   Wikipedia