Wrapped exponential 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 $\operatorname{E}(\theta)$ .

References

Category:Continuous distributions

Category:Directional statistics