In probability theory and statistics, the asymmetric Laplace distribution (ALD) 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.
A random variable has an asymmetric Laplace(m, λ, κ) distribution if its probability density function is[1] [2]
| f(x;m,λ,\kappa)=\left( | λ |
| \kappa+1/\kappa |
\right)
| -(x-m)λs\kappas | |
| e |
where s=sgn(x-m), or alternatively:
f(x;m,λ,\kappa)=
| λ | |
| \kappa+1/\kappa |
\begin{cases} \exp\left((λ/\kappa)(x-m)\right)&ifx<m \\[4pt] \exp(-λ\kappa(x-m))&ifx\geqm \end{cases}
Here, m is a location parameter, λ > 0 is a scale parameter, and κ is an asymmetry parameter. When κ = 1, (x-m)s κs simplifies to |x-m| and the distribution simplifies to the Laplace distribution.
The cumulative distribution function is given by:
F(x;m,λ,\kappa)=\begin{cases}
| \kappa2 | |
| 1+\kappa2 |
\exp((λ/\kappa)(x-m))&ifx\leqm \\[4pt] 1-
| 1 | |
| 1+\kappa2 |
\exp(-λ\kappa(x-m))&ifx>m \end{cases}
The ALD characteristic function is given by:
| \varphi(t;m,λ,\kappa)= | ei | ||||||
|
For m = 0, the ALD is a member of the family of geometric stable distributions with α = 2. It follows that if
\varphi1
\varphi2
| \varphi= | 1 |
| 1/\varphi1+1/\varphi2-1 |
is also an ALD characteristic function with location parameter
m=0
| 1 | = | |
| λ2 |
| 1 | + | |||||
|
| 1 | ||||||
|
and the new skewness parameter κ obeys:
| \kappa2-1 | = | |
| \kappaλ |
| + | |||||||
| \kappa1λ1 |
| |||||||
| \kappa2λ2 |
The n-th moment of the ALD about m is given by
| ||||
| E[(x-m) |
(\kappa-(n+1)-(-\kappa)n+1)
| ||||
| E[x |
\left(
| ||||
| \sum | ||||
| i=0 |
| 1 | |
| (mλ\kappa)i+1 |
| ||||
| -\sum | ||||
| i=0 |
| 1 | |
| (-mλ/\kappa)i+1 |
\right)
| = | λmn+1 |
| \kappa+1/\kappa |
\left(emE-n(mλ\kappa) -e-mE-n(-mλ/\kappa) \right)
where
En()
| n-1 | |
| E | |
| n(x)=x |
\Gamma(1-n,x)
The first moment about zero is the mean:
| \mu=E[x]=m- | \kappa-1/\kappa |
| λ |
The variance is:
\sigma2=E[x2]-\mu
| ||||
and the skewness is:
| E[x3]-3\mu\sigma2-\mu3 | = | |
| \sigma3 |
| 2\left(1-\kappa6\right) | |
| \left(\kappa4+1\right)3/2 |
Asymmetric Laplace variates (X) may be generated from a random variate U drawn from the uniform distribution in the interval (-κ,1/κ) by:
| X=m- | 1 |
| λs\kappas |
log(1-Us\kappaS)
They may also be generated as the difference of two exponential distributions. If X1 is drawn from exponential distribution with mean and rate (m1,λ/κ) and X2 is drawn from an exponential distribution with mean and rate (m2,λκ) then X1 - X2 is distributed according to the asymmetric Laplace distribution with parameters (m1-m2, λ, κ)
The differential entropy of the ALD is
| infty | |
| H=-\int | |
| -infty |
fAL(x)log(fAL(x))dx=1-log\left(
| λ | |
| \kappa+1/\kappa |
\right)
The ALD has the maximum entropy of all distributions with a fixed value (1/λ) of
(x-m)s\kappas
s=sgn(x-m)
An alternative parametrization is made possible by the characteristic function:
| \varphi(t;\mu,\sigma,\beta)= | ei |
| 1-i\beta\sigmat+\sigma2t2 |
where
\mu
\sigma
\beta
f(x;\mu,\sigma,\beta)=
| 1 | |
| 2\sigmaB0 |
\begin{cases} \exp\left(
| x-\mu | |
| \sigmaB- |
\right)&ifx<\mu \\[4pt] \exp(-
| x-\mu | |
| \sigmaB+ |
)&ifx\geq\mu \end{cases}
where
| 2/4} | |
| B | |
| 0=\sqrt{1+\beta |
| \pm=B | |
| B | |
| 0\pm\beta/2 |
B+B-=1,\PB+-B-=\beta
The n-th moment about
\mu
| ||||
| E[(x-\mu) |
((B+)n+1+(-1)n(B-)n+1)
\beta
3\beta/\sqrt{2}
\beta
The Asymmetric Laplace distribution is commonly used with an alternative parameterization for performing quantile regression in a Bayesian inference context.[4] Under this approach, the
\kappa
p
f(x;m,λ,p)=
| p(1-p) | |
| λ |
\begin{cases} \exp\left(-((p-1)/λ)(x-m)\right)&ifx\leqm \\[4pt] \exp(-(p/λ)(x-m))&ifx>m \end{cases}
Where, m is a location parameter, λ > 0 is a scale parameter, and 0 < p < 1 is a percentile parameter.
The mean (
\mu
\sigma2
| \mu=m+ | 1-2p |
| p(1-p) |
λ
| ||||
| \sigma |
λ2
The cumulative distribution function is given[5] by:
F(x;m,λ,p)=\begin{cases} p\exp(
| 1-p | |
| λ |
(x-m))&ifx\leqm \\[4pt] 1-(1-p)\exp(
| -p | |
| λ |
(x-m))&ifx>m \end{cases}
The asymmetric Laplace distribution has applications in finance and neuroscience. For the example in finance, S.G. Kou developed a model for financial instrument prices incorporating an asymmetric Laplace distribution to address problems of skewness, kurtosis and the volatility smile that often occur when using a normal distribution for pricing these instruments.[6] Another example is in neuroscience in which its convolution with normal distribution is considered as a model for brain stopping reaction times.[7]