Wrapped exponential distribution

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Wrapped Exponential
Probability density function

The support is chosen to be [0,2π]
Cumulative distribution function

The support is chosen to be [0,2π]
Parameters \lambda >0
Support 0\leq \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[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\leq \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\leq 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\leq \theta <2\pi for a fixed value of the expectation \operatorname {E}(\theta ).[1]

See also

References

  1. ↑ 1.0 1.1 Jammalamadaka, S. Rao; Kozubowski, Tomasz J. (2004). "New Families of Wrapped Distributions for Modeling Skew Circular Data". Communications in Statistics - Theory and Methods 33 (9): 2059–2074. doi:10.1081/STA-200026570. Retrieved 2011-06-13. 
 
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