Wrapped asymmetric Laplace distribution

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Wrapped asymmetric Laplace distribution
Probability density function
WrappedAsymmetricLaplacePDF.jpg
Wrapped asymmetric Laplace PDF with m = 0.Note that the κ =  2 and 1/2 curves are mirror images about θ=π
Parameters location

scale (real)

asymmetry (real)
Support
PDF(see article)
Mean (circular)
Variance (circular)
CF

In probability theory and directional statistics, a wrapped asymmetric Laplace distribution is a wrapped probability distribution that results from the "wrapping" of the asymmetric Laplace distribution around the unit circle. For the symmetric case (asymmetry parameter κ = 1), the distribution becomes a wrapped Laplace distribution. The distribution of the ratio of two circular variates (Z) from two different wrapped exponential distributions will have a wrapped asymmetric Laplace distribution. These distributions find application in stochastic modelling of financial data.

Definition[edit]

The probability density function of the wrapped asymmetric Laplace distribution is:[1]

where is the asymmetric Laplace distribution. The angular parameter is restricted to . The scale parameter is which is the scale parameter of the unwrapped distribution and is the asymmetry parameter of the unwrapped distribution.

Characteristic function[edit]

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

which yields an alternate expression for the wrapped asymmetric Laplace PDF in terms of the circular variable z=ei(θ-m) valid for all real θ and m:

where is the Lerch transcendent function and coth() is the hyperbolic cotangent function.

Circular moments[edit]

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

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

The circular variance is then 1 − R

Generation of random variates[edit]

If X is a random variate drawn from an asymmetric Laplace distribution (ALD), then will be a circular variate drawn from the wrapped ALD, and, will be an angular variate drawn from the wrapped ALD with .

Since the ALD is the distribution of the difference of two variates drawn from the exponential distribution, it follows that if Z1 is drawn from a wrapped exponential distribution with mean m1 and rate λ/κ and Z2 is drawn from a wrapped exponential distribution with mean m2 and rate λκ, then Z1/Z2 will be a circular variate drawn from the wrapped ALD with parameters ( m1 - m2 , λ, κ) and will be an angular variate drawn from that wrapped ALD with .

See also[edit]

References[edit]

  1. ^ 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.