scale (real) 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.

These distributions find application in stochastic modelling of financial data.

The probability density function of the wrapped asymmetric Laplace distribution is:[1] where

which is the scale parameter of the unwrapped distribution and

is the asymmetry parameter of the unwrapped distribution.

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=ei(θ-m) valid for all real θ and m: where

the circular moments of the wrapped asymmetric Laplace distribution are the characteristic function of the asymmetric Laplace distribution evaluated at integer arguments: The first moment is then the average value of z, also known as the mean resultant, or mean resultant vector: The mean angle is

and the length of the mean resultant is The circular variance is then 1 − R If X is a random variate drawn from an asymmetric Laplace distribution (ALD), then

will be a circular variate drawn from the wrapped ALD, and,

will be an angular variate drawn from the wrapped ALD with

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

will be an angular variate drawn from that wrapped ALD with