Wrapped Cauchy
Probability density function The support is chosen to be [-π,π)
Cumulative distribution function The support is chosen to be [-π,π)
Parameters	${\displaystyle \mu }$ Real ${\displaystyle \gamma >0}$
Support
PDF	${\displaystyle \frac{1}{2\pi }\frac{\sinh (\gamma )}{\cosh (\gamma )-\cos (\theta -\mu )}}$
CDF	${\displaystyle }$
Mean	${\displaystyle \mu }$ (circular)
Variance	${\displaystyle 1-e^{-\gamma }}$ (circular)
Entropy	${\displaystyle \ln (2\pi (1-e^{-2\gamma }))}$ (differential)
CF	${\displaystyle e^{in\mu -|n|\gamma }}$

Inprobability theory anddirectional statistics, awrapped Cauchy distribution is awrapped probability distribution that results from the "wrapping" of theCauchy distribution around theunit circle. The Cauchy distribution is sometimes known as a Lorentzian distribution, and the wrapped Cauchy distribution may sometimes be referred to as a wrapped Lorentzian distribution.

The wrapped Cauchy distribution is often found in the field of spectroscopy where it is used to analyze diffraction patterns (e.g. seeFabry–Pérot interferometer).

Theprobability density function of the wrappedCauchy distribution is:^[1]

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 thecharacteristic function of the Cauchy distribution yields:

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

The PDF may also be expressed in terms of the circular variablez =e^iθ and the complex parameterζ =e^i(μ+iγ)

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

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:

${\displaystyle ⟨z^n⟩={\int }_{\mathrm{\Gamma }}{e}^{in\theta }{f}_{\text{WC}}(\theta ;\mu ,\gamma )d\theta =e^{in\mu -|n|\gamma }.}$

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

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

The mean angle is

${\displaystyle ⟨\theta ⟩=\mathrm{A}\mathrm{r}\mathrm{g}⟨z⟩=\mu }$

and the length of the mean resultant is

${\displaystyle R=|⟨z⟩|=e^{-\gamma }}$

yielding a circular variance of 1 −R.

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A series ofN 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${\displaystyle \gamma }$.

Theinformation entropy of the wrapped Cauchy distribution is defined as:^[1]

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

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 aFourier 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^[2] 4.224.15) and

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

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

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

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 theTaylor expansion for the logarithm of${\displaystyle (1-e^{-2\gamma })}$ so the entropy may be written inclosed form as:

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

IfX is Cauchy distributed with medianμ and scale parameterγ, then the complex variable

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

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

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

where

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

andψ expresses the two parameters of the associated linear Cauchy distribution forx as acomplex 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 inz 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^[3][4] (also the complex Cauchy distribution^[3]) with parametersμ andγ. (See alsoMcCullagh's parametrization of the Cauchy distributions andPoisson 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 integern ≥ 1. For |φ| < 1, the transformation

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

isholomorphic on the unit disk, and the transformed variableU(Z,φ) is distributed as complex Cauchy with parameterU(ζ,φ).

Given a samplez₁, ...,z_n of sizen > 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 withζ⁽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:

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

Forn ≤ 4, closed-form expressions are known for${\displaystyle \hat{\zeta }}$.^[6] The density of the maximum-likelihood estimator att 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 forp₃ andp₄ 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. Retrieved31 Dec 2009.
• Fisher, N. I. (1996).Statistical Analysis of Circular Data.Cambridge University Press.ISBN 978-0-521-56890-6. Retrieved2010-02-09.

• Continuous distributions
• Directional statistics

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