Wrapped asymmetric Laplace distribution

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

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

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

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

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

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:

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

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

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

${\displaystyle \langle z^{n}\rangle =\varphi _{n}(m,\lambda ,\kappa )}$

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

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

The mean angle is ${\displaystyle (-\pi \leq \langle \theta \rangle \leq \pi )}$

${\displaystyle \langle \theta \rangle =\arg(\,\langle z\rangle \,)=\arg(e^{im})}$

and the length of the mean resultant is

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

The circular variance is then 1 − R

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

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 ${\displaystyle \theta =\arg(Z_{1}/Z_{2})+\pi }$ will be an angular variate drawn from that wrapped ALD with ${\displaystyle -\pi <\theta \leq \pi }$.

• Wrapped distribution
• Directional statistics