Wrapped Cauchy distribution

The probability density function of the wrapped Cauchy distribution is:[1]

${\displaystyle f_{WC}(\theta$ ;\mu ,\gamma )=\sum _{n=-\infty }^{\infty }{\frac {\gamma }{\pi (\gamma ^{2}+(\theta -\mu +2\pi n)^{2})}}\qquad -\pi <\theta <\pi }

where ${\displaystyle \gamma }$ is the scale factor and ${\displaystyle \mu }$ is the peak position of the "unwrapped" distribution. Expressing the above pdf in terms of the characteristic function of the Cauchy distribution yields:

${\displaystyle f_{WC}(\theta$ ;\mu ,\gamma )={\frac {1}{2\pi }}\sum _{n=-\infty }^{\infty }e^{in(\theta -\mu )-|n|\gamma }={\frac {1}{2\pi }}\,\,{\frac {\sinh \gamma }{\cosh \gamma -\cos(\theta -\mu )}}}

The PDF may also be expressed in terms of the circular variable z = e^iθ and the complex parameter ζ = e^i(μ+iγ)

${\displaystyle f_{WC}(z;\zeta )={\frac {1}{2\pi }}\,\,{\frac {1-|\zeta |^{2}}{|z-\zeta |^{2}}}}$

where, as shown below, ζ = ⟨z⟩.

In terms of the circular variable ${\displaystyle z=e^{i\theta }}$ the circular moments of the wrapped Cauchy distribution are the characteristic function of the Cauchy distribution evaluated at integer arguments:

${\displaystyle \langle z^{n}\rangle =\int _{\Gamma }e^{in\theta }\,f_{WC}(\theta$ ;\mu ,\gamma )\,d\theta =e^{in\mu -|n|\gamma }.}

where ${\displaystyle \Gamma \,}$ is some interval of length ${\displaystyle 2\pi }$. The first moment is then the average value of z, also known as the mean resultant, or mean resultant vector:

${\displaystyle \langle z\rangle =e^{i\mu -\gamma }}$

The mean angle is

${\displaystyle \langle \theta \rangle =\mathrm {Arg} \langle z\rangle =\mu }$

and the length of the mean resultant is

${\displaystyle R=|\langle z\rangle |=e^{-\gamma }}$

yielding a circular variance of 1 − R.

A series of N measurements ${\displaystyle z_{n}=e^{i\theta _{n}}}$ drawn from a wrapped Cauchy distribution may be used to estimate certain parameters of the distribution. The average of the series ${\displaystyle {\overline {z}}}$ is defined as

${\displaystyle {\overline {z}}={\frac {1}{N}}\sum _{n=1}^{N}z_{n}}$

and its expectation value will be just the first moment:

${\displaystyle \langle {\overline {z}}\rangle =e^{i\mu -\gamma }}$

In other words, ${\displaystyle {\overline {z}}}$ is an unbiased estimator of the first moment. If we assume that the peak position ${\displaystyle \mu }$ lies in the interval ${\displaystyle [-\pi ,\pi )}$, then Arg${\displaystyle ({\overline {z}})}$ will be a (biased) estimator of the peak position ${\displaystyle \mu }$.

Viewing the ${\displaystyle z_{n}}$ as a set of vectors in the complex plane, the ${\displaystyle {\overline {R}}^{2}}$ statistic is the length of the averaged vector:

${\displaystyle {\overline {R}}^{2}={\overline {z}}\,{\overline {z^{*}}}=\left({\frac {1}{N}}\sum _{n=1}^{N}\cos \theta _{n}\right)^{2}+\left({\frac {1}{N}}\sum _{n=1}^{N}\sin \theta _{n}\right)^{2}}$

and its expectation value is

${\displaystyle \langle {\overline {R}}^{2}\rangle ={\frac {1}{N}}+{\frac {N-1}{N}}e^{-2\gamma }.}$

In other words, the statistic

${\displaystyle R_{e}^{2}={\frac {N}{N-1}}\left({\overline {R}}^{2}-{\frac {1}{N}}\right)}$

will be an unbiased estimator of ${\displaystyle e^{-2\gamma }}$, and ${\displaystyle \ln(1/R_{e}^{2})/2}$ will be a (biased) estimator of ${\displaystyle \gamma }$.

The information entropy of the wrapped Cauchy distribution is defined as:[1]

${\displaystyle H=-\int _{\Gamma }f_{WC}(\theta$ ;\mu ,\gamma )\,\ln(f_{WC}(\theta ;\mu ,\gamma ))\,d\theta }

where ${\displaystyle \Gamma }$ is any interval of length ${\displaystyle 2\pi }$. The logarithm of the density of the wrapped Cauchy distribution may be written as a Fourier series in ${\displaystyle \theta \,}$:

${\displaystyle \ln(f_{WC}(\theta$ ;\mu ,\gamma ))=c_{0}+2\sum _{m=1}^{\infty }c_{m}\cos(m\theta )}

where

${\displaystyle c_{m}={\frac {1}{2\pi }}\int _{\Gamma }\ln \left({\frac {\sinh \gamma }{2\pi (\cosh \gamma -\cos \theta )}}\right)\cos(m\theta )\,d\theta }$

which yields:

${\displaystyle c_{0}=\ln \left({\frac {1-e^{-2\gamma }}{2\pi }}\right)}$

(c.f. Gradshteyn and Ryzhik[2] 4.224.15) and

${\displaystyle c_{m}={\frac {e^{-m\gamma }}{m}}\qquad \mathrm {for} \,m>0}$

(c.f. Gradshteyn and Ryzhik[2] 4.397.6). The characteristic function representation for the wrapped Cauchy distribution in the left side of the integral is:

${\displaystyle f_{WC}(\theta$ ;\mu ,\gamma )={\frac {1}{2\pi }}\left(1+2\sum _{n=1}^{\infty }\phi _{n}\cos(n\theta )\right)}

where ${\displaystyle \phi _{n}=e^{-|n|\gamma }}$. Substituting these expressions into the entropy integral, exchanging the order of integration and summation, and using the orthogonality of the cosines, the entropy may be written:

${\displaystyle H=-c_{0}-2\sum _{m=1}^{\infty }\phi _{m}c_{m}=-\ln \left({\frac {1-e^{-2\gamma }}{2\pi }}\right)-2\sum _{m=1}^{\infty }{\frac {e^{-2n\gamma }}{n}}}$

The series is just the Taylor expansion for the logarithm of ${\displaystyle (1-e^{-2\gamma })}$ so the entropy may be written in closed form as:

${\displaystyle H=\ln(2\pi (1-e^{-2\gamma }))\,}$

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

${\displaystyle Z={\frac {X-i}{X+i}}}$

has unit modulus and is distributed on the unit circle with density:[3]

${\displaystyle f_{CC}(\theta ,\mu ,\gamma )={\frac {1}{2\pi }}{\frac {1-|\zeta |^{2}}{|e^{i\theta }-\zeta |^{2}}}}$

where

${\displaystyle \zeta ={\frac {\psi -i}{\psi +i}}}$

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

${\displaystyle \psi =\mu +i\gamma \,}$

It can be seen that the circular Cauchy distribution has the same functional form as the wrapped Cauchy distribution in z and ζ (i.e. f_WC(z,ζ)). The circular Cauchy distribution is a reparameterized wrapped Cauchy distribution:

${\displaystyle f_{CC}(\theta ,m,\gamma )=f_{WC}\left(e^{i\theta },\,{\frac {m+i\gamma -i}{m+i\gamma +i}}\right)}$

The distribution ${\displaystyle f_{CC}(\theta$ ;\mu ,\gamma )} is called the circular Cauchy distribution[3][4] (also the complex Cauchy distribution[3]) with parameters μ and γ. (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

${\displaystyle \operatorname {E} [Z^{n}]=\zeta ^{n},\quad \operatorname {E} [{\bar {Z}}^{n}]={\bar {\zeta }}^{n}}$

for integer n ≥ 1. For |φ| < 1, the transformation

${\displaystyle U(z,\phi )={\frac {z-\phi }{1-{\bar {\phi }}z}}}$

is holomorphic on the unit disk, and the transformed variable U(Z, φ) is distributed as complex Cauchy with parameter U(ζ, φ).

Given a sample z₁, ..., z_n of size n > 2, the maximum-likelihood equation

${\displaystyle n^{-1}U\left(z,{\hat {\zeta }}\right)=n^{-1}\sum U\left(z_{j},{\hat {\zeta }}\right)=0}$

can be solved by a simple fixed-point iteration:

${\displaystyle \zeta ^{(r+1)}=U\left(n^{-1}U(z,\zeta ^{(r)}),\,-\zeta ^{(r)}\right)\,}$

starting with ζ⁽⁰⁾ = 0. The sequence of likelihood values is non-decreasing, and the solution is unique for samples containing at least three distinct values.[5]

The maximum-likelihood estimate for the median (${\displaystyle {\hat {\mu }}}$) and scale parameter (${\displaystyle {\hat {\gamma }}}$) of a real Cauchy sample is obtained by the inverse transformation:

${\displaystyle {\hat {\mu }}\pm i{\hat {\gamma }}=i{\frac {1+{\hat {\zeta }}}{1-{\hat {\zeta }}}}.}$

For n ≤ 4, closed-form expressions are known for ${\displaystyle {\hat {\zeta }}}$.[6] The density of the maximum-likelihood estimator at t in the unit disk is necessarily of the form:

${\displaystyle {\frac {1}{4\pi }}{\frac {p_{n}(\chi (t,\zeta ))}{(1-|t|^{2})^{2}}},}$

where

${\displaystyle \chi (t,\zeta )={\frac {|t-\zeta |^{2}}{4(1-|t|^{2})(1-|\zeta |^{2})}}}$.

Formulae for p₃ and p₄ are available.[7]

• Wrapped distribution
• Dirac comb
• Wrapped normal distribution
• Circular uniform distribution
• McCullagh's parametrization of the Cauchy distributions

• Borradaile, Graham (2003). Statistics of Earth Science Data. Springer. ISBN 978-3-540-43603-4. Retrieved 31 Dec 2009.
• Fisher, N. I. (1996). Statistical Analysis of Circular Data. Cambridge University Press. ISBN 978-0-521-56890-6. Retrieved 2010-02-09.