Wrapped exponential distribution

Parameters Probability density function The support is chosen to be [0,2π] Cumulative distribution function The support is chosen to be [0,2π] $\lambda>0$ $0\le\theta<2\pi$ $\frac{\lambda e^{-\lambda \theta}}{1-e^{-2\pi \lambda}}$ $\frac{1-e^{-\lambda \theta}}{1-e^{-2\pi \lambda}}$ $\arctan(1/\lambda)$ (circular) $1-\frac{\lambda}{\sqrt{1+\lambda^2}}$ (circular) $1+\ln\left(\frac{\beta-1}{\lambda}\right)-\frac{\beta}{\beta-1}\ln(\beta)$ where $\beta=e^{2\pi\lambda}$ (differential) $\frac{1}{1-in/\lambda}$

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[1]

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

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 $0\le X < 2\pi$.

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:

$f_{WE}(\theta;\lambda)=\frac{1}{2\pi}\sum_{n=-\infty}^\infty \frac{e^{in\theta}}{1-in/\lambda} .$

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

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

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 $\operatorname{E}(\theta)$.[1]