Cauchy Distribution - Circular Cauchy Distribution

Circular Cauchy Distribution

If X is Cauchy distributed with median μ and scale parameter γ, then the complex variable

has unit modulus and is distributed on the unit circle with density:

with respect to the angular variable, where

and expresses the two parameters of the associated linear Cauchy distribution for x as a complex number:

The distribution is called the circular Cauchy distribution (also the complex Cauchy distribution) with parameter . The circular Cauchy distribution is related to the wrapped Cauchy distribution. If is a wrapped Cauchy distribution with the parameter representing the parameters of the corresponding "unwrapped" Cauchy distribution in the variable y where, then

See also McCullagh's parametrization of the Cauchy distributions and Poisson kernel for related concepts.

The circular Cauchy distribution expressed in complex form has finite moments of all orders

for integer . For, the transformation

is holomorphic on the unit disk, and the transformed variable is distributed as complex Cauchy with parameter .

Given a sample of size n > 2, the maximum-likelihood equation

can be solved by a simple fixed-point iteration:

starting with The sequence of likelihood values is non-decreasing, and the solution is unique for samples containing at least three distinct values.

The maximum-likelihood estimate for the median and scale parameter of a real Cauchy sample is obtained by the inverse transformation:

For n ≤ 4, closed-form expressions are known for . The density of the maximum-likelihood estimator at t in the unit disk is necessarily of the form:

where

.

Formulae for and are available.