Wrapped asymmetric Laplace distribution

Wrapped asymmetric Laplace distribution

Probability density function Wrapped asymmetric Laplace PDF with m = 0.Note that the κ =  2 and 1/2 curves are mirror images about θ=π
Parameters	$m$ location $( 0 \le m < 2 \pi )$ $\lambda > 0$ scale (real) $\kappa > 0$ asymmetry (real)
Support	$0\le\theta<2\pi$
PDF	(see article)
Mean	$m$ (circular)
Variance	$1-\frac{\lambda ^2}{\sqrt{\left(\frac{1}{\kappa ^2}+\lambda^2 \right) \left(\kappa ^2+\lambda ^2\right)}}$ (circular)
CF	$\frac{\lambda^2 e^{i m n}}{\left(n-i \lambda/\kappa \right) \left(n+i \lambda\kappa \right)}$

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:^[1]

$\begin{align} f_{WAL}(\theta;m,\lambda,\kappa) & =\sum_{k=-\infty}^\infty f_{AL}(\theta+2 \pi k,m,\lambda,\kappa) \\[10pt] & = \begin{cases} \dfrac{e^{-(\theta-m)\lambda\kappa}} {1-e^{-2\pi\kappa\lambda}}- \dfrac{e^{(\theta-m)\lambda/\kappa}} {1-e^{2\pi\lambda/\kappa}} & \text{if } \theta \geq m \\[12pt] \dfrac{e^{-(\theta-m)\lambda\kappa}} {e^{2 \pi \lambda\kappa}-1}- \dfrac{e^{ (\theta-m)\lambda/\kappa}} {e^{-2\pi\lambda/\kappa }-1} & \text{if }\theta<m \end{cases} \end{align}$

where $f_{AL}$ 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.

Characteristic function

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

$\varphi_n(m,\lambda,\kappa)=\frac{\lambda^2 e^{i m n}}{\left(n-i \lambda/\kappa \right) \left(n+i \lambda\kappa \right)}$

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:

$\begin{align} f_{WAL}(z;m,\lambda,\kappa) &= \frac{1}{2\pi}\sum_{n=-\infty}^\infty \varphi_n(0,\lambda,\kappa)z^{-n} \\[10pt] &= \frac{\lambda}{\pi(\kappa+1/\kappa)} \begin{cases} \textrm{Im}\left(\Phi (z,1,-i \lambda\kappa )-\Phi \left(z,1,i \lambda /\kappa \right)\right)-\frac{1}{2 \pi } & \text{if }z \ne 1 \\[12pt] \coth(\pi\lambda\kappa)+\coth(\pi\lambda/\kappa) & \text{if }z=1 \end{cases} \end{align}$

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

Circular moments

In terms of the circular variable $z=e^{i\theta}$ 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:

$\langle z \rangle =\frac{\lambda^2 e^{i m}}{\left(1-i \lambda/\kappa \right) \left(1+i \lambda\kappa \right)}$

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

$\langle \theta \rangle=\arg(\,\langle z \rangle\,)=\arg(e^{i m})$

and the length of the mean resultant is

$R=|\langle z \rangle| = \frac{\lambda ^2}{\sqrt{\left(\frac{1}{\kappa ^2}+\lambda^2 \right) \left(\kappa ^2+\lambda ^2\right)}}.$

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 $Z=e^{i X}$ 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 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 $\theta=\arg(Z_1/Z_2)+\pi$ will be an angular variate drawn from that wrapped ALD with $-\pi<\theta\leq \pi$.

See also

• Wrapped distribution
• Directional statistics

References

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.