McCullagh's parametrization of the Cauchy distributions

In probability theory, the "standard" Cauchy distribution is the probability distribution whose probability density function (pdf) 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 (aθ + b)/(cθ + 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".

Differential equation

McCullagh's parametrization of the pdf of the Cauchy distribution is a solution to the following differential equation:

\left\{{\begin{array}{l}f'(x)\left(\mu ^{2}+\sigma ^{2}+x^{2}-2\mu x\right)+f(x)(2x-2\mu )=0,\\f(0)={\frac {1}{\pi \left|\sigma \right|\left({\frac {\mu ^{2}}{\sigma ^{2}}}+1\right)}}\end{array}}\right\}

References

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

Probability distributions

List

Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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