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]
f W A L ( θ ; m , λ , κ ) = ∑ k = − ∞ ∞ f A L ( θ + 2 π k , m , λ , κ ) = κ λ κ 2 + 1 { e − ( θ − m ) λ κ 1 − e − 2 π λ κ − e ( θ − m ) λ / κ 1 − e 2 π λ / κ if θ ≥ m e − ( θ − m ) λ κ e 2 π λ κ − 1 − e ( θ − m ) λ / κ e − 2 π λ / κ − 1 if θ < m {\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 f A L {\displaystyle f_{AL}} is the asymmetric Laplace distribution . The angular parameter is restricted to 0 ≤ θ < 2 π {\displaystyle 0\leq \theta <2\pi } . The scale parameter is λ > 0 {\displaystyle \lambda >0} which is the scale parameter of the unwrapped distribution and κ > 0 {\displaystyle \kappa >0} is the asymmetry parameter of the unwrapped distribution.
The cumulative distribution function F W A L {\displaystyle F_{WAL}} is therefore:
F W A L ( θ ; m , λ , κ ) = κ λ κ 2 + 1 { e m λ κ ( 1 − e − θ λ κ ) λ κ ( e 2 π λ κ − 1 ) + κ e − m λ / κ ( 1 − e θ λ / κ ) λ ( e − 2 π λ / κ − 1 ) if θ ≤ m 1 − e − ( θ − m ) λ κ λ κ ( 1 − e − 2 π λ κ ) + κ ( 1 − e ( θ − m ) λ / κ ) λ ( 1 − e 2 π λ / κ ) + e m λ κ − 1 λ κ ( e 2 π λ κ − 1 ) + κ ( e − m λ / κ − 1 ) λ ( e − 2 π λ / κ − 1 ) if θ > m {\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}}} Characteristic function [ edit ] The characteristic function of the wrapped asymmetric Laplace is just the characteristic function of the asymmetric Laplace function evaluated at integer arguments:
φ n ( m , λ , κ ) = λ 2 e i m n ( n − i λ / κ ) ( n + i λ κ ) {\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=ei(θ-m) valid for all real θ and m :
f W A L ( z ; m , λ , κ ) = 1 2 π ∑ n = − ∞ ∞ φ n ( 0 , λ , κ ) z − n = λ π ( κ + 1 / κ ) { Im ( Φ ( z , 1 , − i λ κ ) − Φ ( z , 1 , i λ / κ ) ) − 1 2 π if z ≠ 1 coth ( π λ κ ) + coth ( π λ / κ ) if z = 1 {\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 Φ ( ) {\displaystyle \Phi ()} is the Lerch transcendent function and coth() is the hyperbolic cotangent function.
In terms of the circular variable z = e i θ {\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:
⟨ z n ⟩ = φ n ( m , λ , κ ) {\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:
⟨ z ⟩ = λ 2 e i m ( 1 − i λ / κ ) ( 1 + i λ κ ) {\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 ( − π ≤ ⟨ θ ⟩ ≤ π ) {\displaystyle (-\pi \leq \langle \theta \rangle \leq \pi )}
⟨ θ ⟩ = arg ( ⟨ z ⟩ ) = arg ( e i m ) {\displaystyle \langle \theta \rangle =\arg(\,\langle z\rangle \,)=\arg(e^{im})} and the length of the mean resultant is
R = | ⟨ z ⟩ | = λ 2 ( 1 κ 2 + λ 2 ) ( κ 2 + λ 2 ) . {\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
Generation of random variates [ edit ] If X is a random variate drawn from an asymmetric Laplace distribution (ALD), then Z = e i X {\displaystyle Z=e^{iX}} will be a circular variate drawn from the wrapped ALD, and, θ = arg ( Z ) + π {\displaystyle \theta =\arg(Z)+\pi } will be an angular variate drawn from the wrapped ALD with 0 < θ ≤ 2 π {\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 Z 1 is drawn from a wrapped exponential distribution with mean m 1 and rate λ/κ and Z 2 is drawn from a wrapped exponential distribution with mean m 2 and rate λκ , then Z 1 /Z 2 will be a circular variate drawn from the wrapped ALD with parameters ( m 1 - m 2 , λ, κ) and θ = arg ( Z 1 / Z 2 ) + π {\displaystyle \theta =\arg(Z_{1}/Z_{2})+\pi } will be an angular variate drawn from that wrapped ALD with − π < θ ≤ π {\displaystyle -\pi <\theta \leq \pi } .
Discrete univariate
with finite support with infinite support
Continuous univariate
supported on a bounded interval supported on a semi-infinite interval supported on the whole real line with support whose type varies
Mixed univariate
Multivariate (joint) Directional Degenerate and singular Families