Wrapped Cauchy distribution

Estimation of parameters

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:

${\displaystyle ⟨\overline{z}⟩=e^{i\mu -\gamma }}$

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:

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

and its expectation value is

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

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 ${\displaystyle \gamma }$.

Entropy

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

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

where ${\displaystyle \mathrm{\Gamma }}$ 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 ${\displaystyle \theta }$:

${\displaystyle \ln (f_{\text{WC}}(\theta ;\mu ,\gamma ))=c_0+2\sum _{m=1}^{\mathrm{\infty }}{c}_m\cos (m\theta )}$

where

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

which yields:

${\displaystyle c_0=\ln \left(\frac{1-e^{-2\gamma }}{2\pi }\right)}$

(cf. Gradshteyn and Ryzhik 4.224.15) and

${\displaystyle c_m=\frac{e^{-m\gamma }}{m}\mathrm{f}\mathrm{o}\mathrm{r}m>0}$

(cf. Gradshteyn and Ryzhik 4.397.6). The characteristic function representation for the wrapped Cauchy distribution in the left side of the integral is:

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

where ${\displaystyle {\varphi }_n=e^{-|n|\gamma }}$. 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:

${\displaystyle H=-c_0-2\sum _{m=1}^{\mathrm{\infty }}{\varphi }_m{c}_m=-\ln \left(\frac{1-e^{-2\gamma }}{2\pi }\right)-2\sum _{m=1}^{\mathrm{\infty }}\frac{e^{-2n\gamma }}{n}}$

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:

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

Circular Cauchy distribution

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:

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

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:

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

The distribution ${\displaystyle f_{\text{CC}}(\theta ;\mu ,\gamma )}$ is called the circular Cauchy distribution (also the complex Cauchy distribution) 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

${\displaystyle \mathrm{E}[Z^n]={\zeta }^n,\mathrm{E}[{\overline{Z}}^n]={\overline{\zeta }}^n}$

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

${\displaystyle U(z,\varphi )=\frac{z-\varphi }{1-\overline{\varphi }z}}$

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.

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 ${\displaystyle \hat{\zeta }}$. 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.