Wrapped exponential distribution

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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
Support
PDF
CDF
Mean (circular)
Variance (circular)
Entropy where (differential)
CF

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[edit]

The probability density function of the wrapped exponential distribution is[1]

for where 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 .

Characteristic function[edit]

The characteristic function of the wrapped exponential is just the characteristic function of the exponential function evaluated at integer arguments:

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:

where is the Lerch transcendent function.

Circular moments[edit]

In terms of the circular variable the circular moments of the wrapped exponential distribution are the characteristic function of the exponential distribution evaluated at integer arguments:

where is some interval of length . The first moment is then the average value of z, also known as the mean resultant, or mean resultant vector:

The mean angle is

and the length of the mean resultant is

and the variance is then 1-R.

Characterisation[edit]

The wrapped exponential distribution is the maximum entropy probability distribution for distributions restricted to the range for a fixed value of the expectation .[1]

See also[edit]

References[edit]

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