Wrapped exponential distribution

Wrapped Exponential
Probability density function Plot of the wrapped exponential PDF The support is chosen to be [0,2π]
Cumulative distribution function Plot of the wrapped exponential CDF 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)
Variance	$1-\frac{\lambda }{\sqrt{1+{\lambda }^2}}$ (circular)
Entropy	$1+\ln \left(\frac{\beta -1}{\lambda }\right)-\frac{\beta }{\beta -1}\ln (\beta )$ where $\beta =e^{2\pi \lambda }$ (differential)
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

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

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:

${\phi }_n(\lambda )=\frac{1}{1-in/\lambda }$

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:

$\begin{array}{cc}{f}_{\text{WE}}(z;\lambda )& =\frac{1}{2\pi }\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{z^{-n}}{1-in/\lambda }\\ & =\{\begin{array}{cc}\frac{\lambda }{\pi }\text{Im}(\Phi (z,1,-i\lambda ))-\frac{1}{2\pi }& \text{if }z\ne 1\\ \frac{\lambda }{1-e^{-2\pi \lambda }}& \text{if }z=1\end{array}\end{array}$

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:

$⟨z^n⟩=\underset{\Gamma }{\int }{e}^{in\theta }{f}_{\text{WE}}(\theta ;\lambda )d\theta =\frac{1}{1-in/\lambda },$

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:

$⟨z⟩=\frac{1}{1-i/\lambda }.$

The mean angle is

$⟨\theta ⟩=\mathrm{A}\mathrm{r}\mathrm{g}⟨z⟩=\arctan (1/\lambda ),$

and the length of the mean resultant is

$R=|⟨z⟩|=\frac{\lambda }{\sqrt{1+{\lambda }^2}}.$

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 $\mathrm{E}(\theta )$.

See also

• Wrapped distribution
• Directional statistics