Wrapped asymmetric Laplace distribution explained

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]

\begin{align} f _{WAL} (\theta;m, λ ,\kappa) & = \sum ^{inf ty} _{k =-inf ty} f _{AL} (\theta +2 \pi k,m, λ ,\kappa) \\[ 10 pt] & = \dfrac{\kappa λ }{\kappa ^{2+1} \begin{cases}} \dfrac{e ^{-(\theta-m)λ\kappa} } - \dfrac & \text \theta \geq m \\[12pt] \dfrac - \dfrac & \text\theta

where

f _{AL} is the asymmetric Laplace distribution. The angular parameter is restricted to 0 \le \theta < 2 \pi . The scale parameter is λ > 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, λ ,\kappa) = \dfrac{\kappa λ }{\kappa ^{2+1} \begin{cases}} \dfrac{e ^{mλ\kappa} ( 1- e ^{-\thetaλ\kappa} )}{ λ \kappa(e ^{2\piλ\kappa} -1 )} + \dfrac{\kappa e ^{-mλ/\kappa} ( 1- e ^{\thetaλ/\kappa} )}{ λ (e ^{-2\piλ/\kappa} -1 )} & if \theta \leq m\\ \dfrac{ 1- e ^{-(\theta-m)λ\kappa} }+\dfrac+\dfrac+\dfrac &\text \theta > m\end

Characteristic function

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

\varphi _{n(m, λ ,\kappa) = (λ ^{2} e ^{i})/(\left(n - i λ /\kappa \right) \left(n + i λ \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:

\begin{align} f _{WAL} (z;m, λ ,\kappa) & = (1)/(2 \pi) \sum ^{inf ty} _{n =-inf ty} \varphi ^{- n} _{n( 0 , λ ,\kappa)z} \\[ 10 pt] & = (λ)/(\pi(\kappa +1 /\kappa)) \begin{cases} rm{Im}\left(\Phi (z, 1 , - i λ \kappa ) - \Phi \left(z, 1 ,i λ /\kappa \right)\right) - (1)/(2 \pi) & if z \ne 1 \\[ 12 pt] \coth(\pi λ \kappa) + \coth(\pi λ /\kappa) & if z =1 \end{cases} \end{align}

where

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

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

\langle z \rangle = (λ ^{2} e ^{i})/(\left( 1- i λ /\kappa \right) \left( 1+ i λ \kappa \right))

The mean angle is

( - \pi \le \langle \theta \rangle \leq \pi) \langle \theta \rangle = \arg(\langle z \rangle) = \arg(e ^{i} )

and the length of the mean resultant is

R = |\langle z \rangle| = (λ ^{2})/(\sqrt{\left( (1)/(+λ ^{2} \right) \left(\kappa ^{2+λ} ^{2\right)} \kappa ^{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 ^{i} 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\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 &lambda;/κ and Z₂ is drawn from a wrapped exponential distribution with mean m₂ and rate &lambda;κ, then Z₁/Z₂ will be a circular variate drawn from the wrapped ALD with parameters (m₁ - m₂, &lambda;, κ) and

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

See also

• Wrapped distribution
• Directional statistics

Notes and References

1. Jammalamadaka . S. Rao . Kozubowski . Tomasz J. . 2004 . New Families of Wrapped Distributions for Modeling Skew Circular Data . Communications in Statistics – Theory and Methods . 33 . 9 . 2059–2074 . 10.1081/STA-200026570. 2011-06-13 .

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