wrapped Lévy distribution

The pdf of the wrapped Lévy distribution is

:$$

f_{WL}(\theta;\mu,c)=\sum_{n=-\infty}^\infty \sqrt{\frac{c}{2\pi}}\,\frac{e^{-c/2(\theta+2\pi n-\mu)}}{(\theta+2\pi n-\mu)^{3/2}}

where the value of the summand is taken to be zero when $\backslash theta+2\backslash pi\; n-\backslash mu\; \backslash le\; 0$, $c$ is the scale factor and $\backslash mu$ is the location parameter. Expressing the above pdf in terms of the characteristic function of the Lévy distribution yields:

:$$

f_{WL}(\theta;\mu,c)=\frac{1}{2\pi}\sum_{n=-\infty}^\infty e^{-in(\theta-\mu)-\sqrt{c|n|}\,(1-i\sgn{n})}=\frac{1}{2\pi}\left(1 + 2\sum_{n=1}^\infty e^{-\sqrt{cn}}\cos\left(n(\theta-\mu) - \sqrt{cn}\,\right)\right)

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

:$\backslash langle\; z^n\backslash rangle=\backslash int\_\backslash Gamma\; e^\{in\backslash theta\}\backslash ,f\_\{WL\}(\backslash theta;\backslash mu,c)\backslash ,d\backslash theta\; =\; e^\{i\; n\; \backslash mu-\backslash sqrt\{c|n|\}\backslash ,(1-i\backslash sgn(n))\}.$

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

:$$

\langle z \rangle=e^{i\mu-\sqrt{c}(1-i)}

The mean angle is

:$$

\theta_\mu=\mathrm{Arg}\langle z \rangle = \mu+\sqrt{c}

and the length of the mean resultant is

:$$

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

• Wrapped distribution
• Directional statistics

• {{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}}

{{DEFAULTSORT:Wrapped Levy distribution}}

Category:Continuous distributions

Category:Directional statistics