Wrapped asymmetric Laplace distribution

Template:Probability 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.

Definition

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

$\begin{array}{cc}{f}_{WAL}(\theta ;m,\lambda ,\kappa )& =\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{f}_{AL}(\theta +2\pi k,m,\lambda ,\kappa )\\ & =\frac{\kappa \lambda }{{\kappa }^2+1}\{\begin{array}{cc}\frac{e^{-(\theta -m)\lambda \kappa }}{1-e^{-2\pi \lambda \kappa }}-\frac{e^{(\theta -m)\lambda /\kappa }}{1-e^{2\pi \lambda /\kappa }}& \text{if }\theta \ge m\\ \frac{e^{-(\theta -m)\lambda \kappa }}{e^{2\pi \lambda \kappa }-1}-\frac{e^{(\theta -m)\lambda /\kappa }}{e^{-2\pi \lambda /\kappa }-1}& \text{if }\theta <m\end{array}\end{array}$

where $f_{AL}$ is the asymmetric Laplace distribution. The angular parameter is restricted to $0\le \theta <2\pi$. The scale parameter is $\lambda >0$ 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:

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

Characteristic function

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

${\phi }_n(m,\lambda ,\kappa )=\frac{{\lambda }^2{e}^{imn}}{(n-i\lambda /\kappa )(n+i\lambda \kappa )}$

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:

$\begin{array}{cc}{f}_{WAL}(z;m,\lambda ,\kappa )& =\frac{1}{2\pi }\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{\phi }_n(0,\lambda ,\kappa )z^{-n}\\ & =\frac{\lambda }{\pi (\kappa +1/\kappa )}\{\begin{array}{cc}\text{Im}(\mathrm{\Phi }(z,1,-i\lambda \kappa )-\mathrm{\Phi }(z,1,i\lambda /\kappa ))-\frac{1}{2\pi }& \text{if }z\ne 1\\ \coth (\pi \lambda \kappa )+\coth (\pi \lambda /\kappa )& \text{if }z=1\end{array}\end{array}$

where $\mathrm{\Phi }()$ 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:

$⟨z^n⟩={\phi }_n(m,\lambda ,\kappa )$

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

$⟨z⟩=\frac{{\lambda }^2{e}^{im}}{(1-i\lambda /\kappa )(1+i\lambda \kappa )}$

The mean angle is $(-\pi \le ⟨\theta ⟩\le \pi )$

$⟨\theta ⟩=\arg (⟨z⟩)=\arg (e^{im})$

and the length of the mean resultant is

$R=|⟨z⟩|=\frac{{\lambda }^2}{\sqrt{(\frac{1}{{\kappa }^2}+{\lambda }^2)({\kappa }^2+{\lambda }^2)}}.$

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 $Z=e^{iX}$ 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 $0<\theta \le 2\pi$.

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 $\theta =\arg (Z_1/Z_2)+\pi$ will be an angular variate drawn from that wrapped ALD with $-\pi <\theta \le \pi$.

See also

• Wrapped distribution
• Directional statistics