wrapped Cauchy distribution

The probability density function of the wrapped Cauchy distribution is:{{cite book |title=Directional Statistics |last=Mardia |first=Kantilal |author-link=Kantilal Mardia |author2=Jupp, Peter E. |year=1999 |publisher=Wiley |isbn=978-0-471-95333-3 }}

:$$

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

:$$

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)

:$$

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

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

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

:$$

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

The mean angle is

:$$

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

and the length of the mean resultant is

:$$

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

yielding a circular variance of 1 − R.

{{Unreferenced section|date=July 2022}}

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

:$\backslash overline\{z\}=\backslash frac\{1\}\{N\}\backslash sum\_\{n=1\}^N\; z\_n$

and its expectation value will be just the first moment:

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

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

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

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

and its expectation value is

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

In other words, the statistic

:$R\_e^2=\backslash frac\{N\}\{N-1\}\backslash left(\backslash overline\{R\}^2-\backslash frac\{1\}\{N\}\backslash right)$

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

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

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

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

:$\backslash ln(f\_\{WC\}(\backslash theta;\backslash mu,\backslash gamma))=c\_0+2\backslash sum\_\{m=1\}^\backslash infty\; c\_m\; \backslash cos(m\backslash theta)$

where

:$c\_m=\backslash frac\{1\}\{2\backslash pi\}\backslash int\_\backslash Gamma\; \backslash ln\backslash left(\backslash frac\{\backslash sinh\backslash gamma\}\{2\backslash pi(\backslash cosh\backslash gamma-\backslash cos\backslash theta)\}\backslash right)\backslash cos(m\; \backslash theta)\backslash ,d\backslash theta$

which yields:

:$c\_0\; =\; \backslash ln\backslash left(\backslash frac\{1-e^\{-2\backslash gamma\}\}\{2\backslash pi\}\backslash right)$

(cf. Gradshteyn and Ryzhik{{cite book |author-first1=Izrail Solomonovich |author-last1=Gradshteyn |author-link1=Izrail Solomonovich Gradshteyn |author-first2=Iosif Moiseevich |author-last2=Ryzhik |author-link2=Iosif Moiseevich Ryzhik |author-first3=Yuri Veniaminovich |author-last3=Geronimus |author-link3=Yuri Veniaminovich Geronimus |author-first4=Michail Yulyevich |author-last4=Tseytlin |author-link4=Michail Yulyevich Tseytlin |editor-first1=Alan |editor-last1=Jeffrey |editor-first2=Daniel |editor-last2=Zwillinger |translator=Scripta Technica, Inc. |title=Table of Integrals, Series, and Products |publisher=Academic Press, Inc. |date=February 2007 |edition=7 |language=en |isbn=0-12-373637-4 |lccn=2010481177 |title-link=Gradshteyn and Ryzhik}} 4.224.15) and

:$c\_m=\backslash frac\{e^\{-m\backslash gamma\}\}\{m\}\backslash qquad\; \backslash mathrm\{for\}\backslash ,m>0$

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

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

where $\backslash phi\_n=\; e^\{-|n|\backslash gamma\}$. 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:

:$H\; =\; -c\_0-2\backslash sum\_\{m=1\}^\backslash infty\; \backslash phi\_m\; c\_m\; =\; -\backslash ln\backslash left(\backslash frac\{1-e^\{-2\backslash gamma\}\}\{2\backslash pi\}\backslash right)-2\backslash sum\_\{m=1\}^\backslash infty\; \backslash frac\{e^\{-2n\backslash gamma\}\}\{n\}$

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

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

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

:$Z\; =\; \backslash frac\{X\; -\; i\}\{X+i\}$

has unit modulus and is distributed on the unit circle with density:{{cite journal |last=McCullagh |first=Peter |date=June 1992 |title=Conditional inference and Cauchy models |url=http://www.stat.uchicago.edu/~pmcc/pubs/paper18.pdf |journal=Biometrika |volume=79 |issue=2 |pages=247–259 |access-date=26 January 2016 |doi=10.1093/biomet/79.2.247}}

:$f\_\{CC\}(\backslash theta,\backslash mu,\backslash gamma)=\; \backslash frac\{1\}\{2\backslash pi\; \}\; \backslash frac\{1\; -\; |\backslash zeta|^2\}\{|e^\{i\; \backslash theta\}\; -\; \backslash zeta|^2\}$

where

:$\backslash zeta\; =\; \backslash frac\{\backslash psi\; -\; i\}\{\backslash psi\; +\; i\}$

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

:$\backslash psi=\backslash mu+i\backslash gamma\backslash ,$

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:

:$$

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

The distribution $f\_\{CC\}(\backslash theta;\; \backslash mu,\backslash gamma)$ is called the circular Cauchy distribution{{cite book |author = K.V. Mardia |year=1972 |title=Statistics of Directional Data |publisher=Academic Press}}{{Page needed|date=November 2010}} (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

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

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

:$U(z,\; \backslash phi)\; =\; \backslash frac\{z\; -\; \backslash phi\}\{1\; -\; \backslash bar\; \backslash phi\; z\}$

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

:$n^\{-1\}\; U\; \backslash left(z,\; \backslash hat\backslash zeta\; \backslash right)\; =\; n^\{-1\}\; \backslash sum\; U\; \backslash left(z\_j,\; \backslash hat\backslash zeta\; \backslash right)\; =\; 0$

can be solved by a simple fixed-point iteration:

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

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.{{cite journal |author=J. Copas |year=1975 |title= On the unimodality of the likelihood function for the Cauchy distribution |journal=Biometrika |volume=62 |issue=3 |pages=701–704 |doi=10.1093/biomet/62.3.701}}

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

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

For n ≤ 4, closed-form expressions are known for $\backslash hat\backslash zeta$.{{cite journal|last1=Ferguson|first1=Thomas S.|author-link= Thomas S. Ferguson |year=1978 |journal=Journal of the American Statistical Association |volume=73|issue=361|title=Maximum Likelihood Estimates of the Parameters of the Cauchy Distribution for Samples of Size 3 and 4|pages=211–213|jstor=2286549 |doi=10.1080/01621459.1978.10480031}} The density of the maximum-likelihood estimator at t in the unit disk is necessarily of the form:

:$\backslash frac\{1\}\{4\backslash pi\}\backslash frac\{p\_n(\backslash chi(t,\; \backslash zeta))\}\{(1\; -\; |t|^2)^2\}\; ,$

where

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

Formulae for p₃ and p₄ are available.{{cite journal |author=P. McCullagh |year=1996 |title=Möbius transformation and Cauchy parameter estimation. |journal=Annals of Statistics |volume=24 |issue=2 |pages=786–808 |doi=10.1214/aos/1032894465 |jstor = 2242674 }}

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

• {{cite book |title=Statistics of Earth Science Data |last=Borradaile |first=Graham |year=2003 |publisher=Springer |isbn=978-3-540-43603-4 |url=https://books.google.com/books?id=R3GpDglVOSEC |access-date=31 Dec 2009}}
• {{cite book |title=Statistical Analysis of Circular Data |last=Fisher |first=N. I. |year=1996 |publisher=Cambridge University Press |isbn=978-0-521-56890-6 |url=https://books.google.com/books?id=IIpeevaNH88C&q=%22circular+variance%22+fisher |access-date=2010-02-09}}

{{ProbDistributions|directional}}

Category:Continuous distributions

Category:Directional statistics