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:

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

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

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

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:

where ${\displaystyle \Phi ()}$ is the Lerch transcendent function and coth() is the hyperbolic cotangent function.

In terms of the circular variable ${\displaystyle z=e^{i\theta }}$ the circular moments of the wrapped asymmetric Laplace distribution are the characteristic function of the asymmetric Laplace distribution evaluated at integer arguments: