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 &kappa; = 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{align} f_{WAL}(\theta;m,\lambda,\kappa) & =\sum_{k=-\infty}^\infty f_{AL}(\theta+2 \pi k,m,\lambda,\kappa) \\[10pt] & = \dfrac{\kappa\lambda}{\kappa^2+1} \begin{cases} \dfrac{e^{-(\theta-m)\lambda\kappa}} {1-e^{-2\pi\lambda\kappa}}- \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 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 $$F_{WAL}$$ is therefore:



F_{WAL}(\theta;m,\lambda,\kappa)=\dfrac{\kappa\lambda}{\kappa^2+1} \begin{cases} \dfrac{e^{m\lambda\kappa}(1-e^{-\theta\lambda\kappa})}{\lambda\kappa(e^{2\pi\lambda\kappa}-1)}+\dfrac{\kappa e^{-m\lambda/\kappa}(1-e^{\theta\lambda/\kappa})}{\lambda(e^{-2\pi\lambda/\kappa}-1)} & \text{if }\theta \leq m\\ \dfrac{1-e^{-(\theta-m)\lambda\kappa}}{\lambda\kappa(1-e^{-2\pi\lambda\kappa})}+\dfrac{\kappa(1-e^{(\theta-m)\lambda/\kappa})}{\lambda(1-e^{2\pi\lambda/\kappa})}+\dfrac{e^{m\lambda\kappa}-1}{\lambda\kappa(e^{2\pi\lambda\kappa}-1)}+\dfrac{\kappa(e^{-m\lambda/\kappa}-1)}{\lambda(e^{-2\pi\lambda/\kappa}-1)} &\text{if } \theta > m \end{cases} $$

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=ei(&theta;-m) valid for all real &theta; 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  &lambda;/&kappa; and Z2 is drawn from a wrapped exponential distribution with mean  m2 and rate  &lambda;&kappa;, then Z1/Z2 will be a circular variate drawn from the wrapped ALD with parameters ( m1 - m2, &lambda;, &kappa;) and $$\theta=\arg(Z_1/Z_2)+\pi$$ will be an angular variate drawn from that wrapped ALD with $$-\pi<\theta\leq  \pi$$.