Wrapped asymmetric Laplace distribution

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

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

:$\begin f_(\theta;m,\lambda,\kappa) & =\sum_^\infty f_(\theta+2 \pi k,m,\lambda,\kappa) \\$0pt& = \dfrac
\begin
\dfrac -
\dfrac
& \text \theta \geq m
\\ 2pt \dfrac -
\dfrac
& \text\theta

where $f_$ is the asymmetric Laplace distribution. The angular parameter is restricted to $0 \le \theta < 2\pi$. The scale parameter is $\lambda > 0$ which is the scale parameter of the unwrapped distribution and $\kappa > 0$ is the asymmetry parameter of the unwrapped distribution.

The cumulative distribution function $F_$ is therefore:

:$F_(\theta;m,\lambda,\kappa)=\dfrac \begin \dfrac+\dfrac & \text\theta \leq m\\ \dfrac+\dfrac+\dfrac+\dfrac &\text \theta > m \end$

Characteristic function

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

:$\varphi_n(m,\lambda,\kappa)=\frac$

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

:$\begin f_(z;m,\lambda,\kappa) &= \frac\sum_^\infty \varphi_n(0,\lambda,\kappa)z^ \\$0pt&= \frac \begin
\textrm\left(\Phi (z,1,-i \lambda\kappa )-\Phi \left(z,1,i \lambda /\kappa \right)\right)-\frac
& \textz \ne 1
\\ 2pt \coth(\pi\lambda\kappa)+\coth(\pi\lambda/\kappa)
& \textz=1
\end
\end

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

Circular moments

In terms of the circular variable $z=e^$ the circular moments of the wrapped asymmetric Laplace distribution are the characteristic function of the asymmetric Laplace distribution evaluated at integer arguments:

:$\langle z^n\rangle=\varphi_n(m,\lambda,\kappa)$

The first moment is then the average value of ''z'', also known as the mean resultant, or mean resultant vector:

:$\langle z \rangle =\frac$

The mean angle is $(-\pi \le \langle \theta \rangle \leq \pi)$

:$\langle \theta \rangle=\arg(\,\langle z \rangle\,)=\arg(e^)$

and the length of the mean resultant is

:$R=, \langle z \rangle, = \frac.$

The circular variance is then 1 − ''R''

Generation of random variates

If X is a random variate drawn from an asymmetric Laplace distribution (ALD), then $Z=e^$ will be a circular variate drawn from the wrapped ALD, and, $\theta=\arg(Z)+\pi$ will be an angular variate drawn from the wrapped ALD with $0<\theta\leq 2 \pi$.

Since the ALD is the distribution of the difference of two variates drawn from the exponential distribution, it follows that if ''Z''₁ is drawn from a wrapped exponential distribution with mean ''m''₁ and rate ''λ/κ'' and ''Z''₂ is drawn from a wrapped exponential distribution with mean ''m''₂ and rate ''λκ'', then ''Z''₁/''Z''₂ will be a circular variate drawn from the wrapped ALD with parameters ( ''m''₁ - ''m''₂ , λ, κ) and $\theta=\arg(Z_1/Z_2)+\pi$ will be an angular variate drawn from that wrapped ALD with $-\pi<\theta\leq \pi$.

See also

* Wrapped distribution
* Directional statistics

References

{{ProbDistributions, directional

Continuous distributions
Directional statistics