Wrapped exponential distribution

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



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 &lambda; 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=ei (&theta;-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 $$\operatorname{E}(\theta)$$.