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]
![{\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))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33a7515212e27b41059ddb3229cb436bf7683569)
where
is the asymmetric Laplace distribution. The angular parameter is restricted to
. The scale parameter is
which is the scale parameter of the unwrapped distribution and
is the asymmetry parameter of the unwrapped distribution.
The cumulative distribution function
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))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33e8038119b959e6119beac3010dadd99d1d9f97)
Characteristic function
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)))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/70ca0b0dfc8e6fcce8d1fb2ffaad45084749e754)
which yields an alternate expression for the wrapped asymmetric Laplace PDF in terms of the circular variable z=ei(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))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db36124eb2b7b7db84902cf00ec585545e61b700)
where
is the Lerch transcendent function and coth() is the hyperbolic cotangent function.
Circular moments
In terms of the circular variable
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 )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e9eb8ba05a4811958bc1bf681aa4e89afa849530)
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)))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7310baa775903a0bf39108e282781c07a2644553)
The mean angle is
![{\displaystyle \langle \theta \rangle =\arg(\,\langle z\rangle \,)=\arg(e^{im})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2fdf6510a34adf3a18927b3db5a76a381f3f8149)
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)))}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a493ac6310fc8b2c25a50eff15350010854dbcb)
The circular variance is then 1 − R