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.

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

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

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

The cumulative distribution function ${\displaystyle F_{WAL}}${\displaystyle F_{WAL}} is therefore:

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

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

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

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

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

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

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

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

and the length of the mean resultant is

${\displaystyle R=|\langle z\rangle |={\frac {\lambda ^{2}}{\sqrt {\left({\frac {1}{\kappa ^{2}}}+\lambda ^{2}\right)\left(\kappa ^{2}+\lambda ^{2}\right)}}}.}${\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 ${\displaystyle Z=e^{iX}}${\displaystyle Z=e^{iX}} will be a circular variate drawn from the wrapped ALD, and, ${\displaystyle \theta =\arg(Z)+\pi }${\displaystyle \theta =\arg(Z)+\pi } will be an angular variate drawn from the wrapped ALD with ${\displaystyle 0<\theta \leq 2\pi }${\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 ${\displaystyle \theta =\arg(Z_{1}/Z_{2})+\pi }${\displaystyle \theta =\arg(Z_{1}/Z_{2})+\pi } will be an angular variate drawn from that wrapped ALD with ${\displaystyle -\pi <\theta \leq \pi }${\displaystyle -\pi <\theta \leq \pi }.

• Wrapped distribution
• Directional statistics

• Continuous distributions
• Directional statistics