Asymmetric Laplace distribution
In probability theory and statistics, the asymmetric Laplace distribution is a continuous probability distribution which is a generalization of the Laplace distribution. Just as the Laplace distribution consists of two exponential distributions of equal scale back-to-back about x = m, the asymmetric Laplace consists of two exponential distributions of unequal scale back to back about x = m, adjusted to assure continuity and normalization. The difference of two variates exponentially distributed with different means and rate parameters will be distributed according to the ALD. When the two rate parameters are equal, the difference will be distributed according to the Laplace distribution.
Characterization
Probability density function
A random variable has an asymmetric Laplace distribution if its probability density function iswhere s=sgn, or alternatively:
Here, m is a location parameter, λ > 0 is a scale parameter, and κ is an asymmetry parameter. When κ = 1, s κs simplifies to |x-m| and the distribution simplifies to the Laplace distribution.
Cumulative distribution function
The cumulative distribution function is given by:Characteristic function
The ALD characteristic function is given by:For m = 0, the ALD is a member of the family of geometric stable distributions with α = 2. It follows that if and are two distinct ALD characteristic functions with m = 0, then
is also an ALD characteristic function with location parameter. The new scale parameter λ obeys
and the new skewness parameter κ obeys:
Moments, mean, variance, skewness
The n-th moment of the ALD about m is given byFrom the binomial theorem, the n-th moment about zero is then:
where is the generalized exponential integral function
The first moment about zero is the mean:
The variance is:
and the skewness is:
Generating asymmetric Laplace variates
Asymmetric Laplace variates may be generated from a random variate U drawn from the uniform distribution in the interval by:where s=sgn.
They may also be generated as the difference of two exponential distributions. If X1 is drawn from exponential distribution with mean and rate and X2 is drawn from an exponential distribution with mean and rate then X1 - X2 is distributed according to the asymmetric Laplace distribution with parameters
Entropy
The differential entropy of the ALD isThe ALD has the maximum entropy of all distributions with a fixed value of where.
Alternative parametrization
An alternative parametrization is made possible by the characteristic function:where is a location parameter, is a scale parameter, is an asymmetry parameter. This is specified in Section 2.6.1 and Section 3.1 of Lihn.
Its probability density function is
where and. It follows that.
The n-th moment about is given by
The mean about zero is:The variance is:The skewness is:The excess kurtosis is:For small, the skewness is about . Thus represents skewness in an almost direct way.
Alternative parameterization for Bayesian quantile regression
The Asymmetric Laplace distribution is commonly used with an alternative parameterization for performing quantile regression in a Bayesian inference context. Under this approach, the parameter describing asymmetry is replaced with a parameter indicating the percentile or quantile desired. Using this parameterization, the likelihood of the Asymmetric Laplace Distribution is equivalent to the loss function employed in quantile regression. With this alternative parameterization, the probability density function is defined as:Where, m is a location parameter, λ > 0 is a scale parameter, and 0 < p < 1 is a percentile parameter.
The mean and variance are calculated as:
The cumulative distribution function is given by: