Wrapped exponential distribution

{{Short description|Probability distribution}}

{{Probability distribution|

name =Wrapped Exponential|

type =density|

pdf_image =Image:WrappedExponentialPDF.png
The support is chosen to be [0,2π]|

cdf_image =Image:WrappedExponentialCDF.png
The support is chosen to be [0,2π]|

parameters =\lambda>0|

support =0\le\theta<2\pi|

pdf =\frac{\lambda e^{-\lambda \theta}}{1-e^{-2\pi \lambda}}|

cdf =\frac{1-e^{-\lambda \theta}}{1-e^{-2\pi \lambda}}|

mean =\arctan(1/\lambda) (circular)|

median =|

mode =|

variance =1-\frac{\lambda}{\sqrt{1+\lambda^2}} (circular)|

skewness =|

kurtosis =|

entropy =1+\ln\left(\frac{\beta-1}{\lambda}\right)-\frac{\beta}{\beta-1}\ln(\beta) where \beta=e^{2\pi\lambda} (differential)|

mgf =|

cf =\frac{1}{1-in/\lambda}|

}}

In probability theory and directional statistics, a wrapped exponential distribution is a wrapped probability distribution that results from the "wrapping" of the exponential distribution around the unit circle.

Definition

The probability density function of the wrapped exponential distribution is{{cite journal |last1=Jammalamadaka |first1=S. Rao |last2=Kozubowski |first2=Tomasz J. |year=2004 |title=New Families of Wrapped Distributions for Modeling Skew Circular Data |journal=Communications in Statistics - Theory and Methods |volume=33 |issue=9 |pages=2059–2074 |url=http://www.pstat.ucsb.edu/faculty/jammalam/html/Some%20Publications/2004_WrappedSkewFamilies_Comm..pdf | doi=10.1081/STA-200026570|access-date=2011-06-13 }}

:

f_{WE}(\theta;\lambda)=\sum_{k=0}^\infty \lambda e^{-\lambda (\theta+2 \pi k)}=\frac{\lambda e^{-\lambda \theta}}{1-e^{-2\pi \lambda}} ,

for 0 \le \theta < 2\pi where \lambda > 0 is the rate parameter of the unwrapped distribution. This is identical to the truncated distribution obtained by restricting observed values X from the exponential distribution with rate parameter λ to the range 0\le X < 2\pi. Note that this distribution is not periodic.

Characteristic function

The characteristic function of the wrapped exponential is just the characteristic function of the exponential function evaluated at integer arguments:

:\varphi_n(\lambda)=\frac{1}{1-in/\lambda}

which yields an alternate expression for the wrapped exponential PDF in terms of the circular variable z=e i (θ-m) valid for all real θ and m:

:

\begin{align}

f_{WE}(z;\lambda)

& =\frac{1}{2\pi}\sum_{n=-\infty}^\infty \frac{z^{-n}}{1-in/\lambda}\\[10pt]

& = \begin{cases}

\frac{\lambda}{\pi}\,\textrm{Im}(\Phi(z,1,-i\lambda))-\frac{1}{2\pi}

& \text{if }z \neq 1

\\[12pt]

\frac{\lambda}{1-e^{-2\pi\lambda}}

& \text{if }z=1

\end{cases}

\end{align}

where \Phi() is the Lerch transcendent function.

Circular moments

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

:\langle z^n\rangle=\int_\Gamma e^{in\theta}\,f_{WE}(\theta;\lambda)\,d\theta = \frac{1}{1-in/\lambda} ,

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

:

\langle z \rangle=\frac{1}{1-i/\lambda} .

The mean angle is

:

\langle \theta \rangle=\mathrm{Arg}\langle z \rangle = \arctan(1/\lambda) ,

and the length of the mean resultant is

:

R=|\langle z \rangle| = \frac{\lambda}{\sqrt{1+\lambda^2}} .

and the variance is then 1-R.

Characterisation

The wrapped exponential distribution is the maximum entropy probability distribution for distributions restricted to the range 0\le \theta < 2\pi for a fixed value of the expectation \operatorname{E}(\theta).

See also

References