McCullagh's parametrization of the Cauchy distributions

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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:

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".

Differential equation[edit]

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

References[edit]