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Milds Asymmetric Laplace distribution

Parameters Probability density function Asymmetric Laplace PDF with m = 0. Note that the κ =  2 and 1/2 curves are mirror images Cumulative distribution function Asymmetric Laplace CDF with m = 0. $m$ location (real) $\lambda >0$ scale (real) $\kappa >0$ asymmetry (real) $x\in (-\infty ;+\infty )\,$ (see article) (see article) $m+{\frac {1-\kappa ^{2}}{\lambda \kappa }}$ $m+(\kappa /\lambda )\log \left({\frac {1+\kappa ^{2}}{2\kappa ^{2}}}\right)$ ${\frac {1+\kappa ^{4}}{\lambda ^{2}\kappa ^{2}}}$ ${\frac {2\left(1-\kappa ^{6}\right)}{\left(\kappa ^{4}+1\right)^{3/2}}}$ ${\frac {6(1+\kappa ^{8})}{(1+\kappa ^{4})^{2}}}$ $\log \left(e\,{\frac {1+\kappa ^{2}}{\kappa \lambda }}\right)$ ${\frac {e^{imt}}{(1+{\frac {it\kappa }{\lambda }})(1-{\frac {it}{\kappa \lambda }})}}$ 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.

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Characterization

Probability density function

A random variable has an asymmetric Laplace(m, λ, κ) distribution if its probability density function is

$f(x;m,\lambda ,\kappa )=\left({\frac {\lambda }{\kappa +1/\kappa }}\right)\,e^{-(x-m)\lambda \,s\kappa ^{s}}$ where s=sgn(x-m), or alternatively:

$f(x;m,\lambda ,\kappa )={\frac {\lambda }{\kappa +1/\kappa }}{\begin{cases}\exp \left((\lambda /\kappa )(x-m)\right)&{\text{if }}x 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.

Cumulative distribution function

The cumulative distribution function is given by:

$F(x;m,\lambda ,\kappa )={\begin{cases}{\frac {\kappa ^{2}}{1+\kappa ^{2}}}\exp((\lambda /\kappa )(x-m))&{\text{if }}x\leq m\\[4pt]1-{\frac {1}{1+\kappa ^{2}}}\exp(-\lambda \kappa (x-m))&{\text{if }}x>m\end{cases}}$ Characteristic function

The ALD characteristic function is given by:

$\varphi (t;m,\lambda ,\kappa )={\frac {e^{imt}}{(1+{\frac {it\kappa }{\lambda }})(1-{\frac {it}{\kappa \lambda }})}}$ For m = 0, the ALD is a member of the family of geometric stable distributions with α = 2. It follows that if $\varphi _{1}$ and $\varphi _{2}$ are two distinct ALD characteristic functions with m = 0, then

$\varphi ={\frac {1}{1/\varphi _{1}+1/\varphi _{2}-1}}$ is also an ALD characteristic function with location parameter $m=0$ . The new scale parameter λ obeys

${\frac {1}{\lambda ^{2}}}={\frac {1}{\lambda _{1}^{2}}}+{\frac {1}{\lambda _{2}^{2}}}$ and the new skewness parameter κ obeys:

${\frac {\kappa ^{2}-1}{\kappa \lambda }}={\frac {\kappa _{1}^{2}-1}{\kappa _{1}\lambda _{1}}}+{\frac {\kappa _{2}^{2}-1}{\kappa _{2}\lambda _{2}}}$ Moments, mean, variance, skewness

The n-th moment of the ALD about m is given by

$E[(x-m)^{n}]={\frac {n!}{\lambda ^{n}(\kappa +1/\kappa )}}\,(\kappa ^{-(n+1)}-(-\kappa )^{n+1})$ From the binomial theorem, the n-th moment about zero (for m not zero) is then:

$E[x^{n}]={\frac {\lambda \,m^{n+1}}{\kappa +1/\kappa }}\,\left(\sum _{i=0}^{n}{\frac {n!}{(n-i)!}}\,{\frac {1}{(m\lambda \kappa )^{i+1}}}-\sum _{i=0}^{n}{\frac {n!}{(n-i)!}}\,{\frac {1}{(-m\lambda /\kappa )^{i+1}}}\right)$ $={\frac {\lambda \,m^{n+1}}{\kappa +1/\kappa }}\left(e^{m\lambda \kappa }E_{-n}(m\lambda \kappa )-e^{-m\lambda /\kappa }E_{-n}(-m\lambda /\kappa )\right)$ where $E_{n}()$ is the generalized exponential integral function $E_{n}(x)=x^{n-1}\Gamma (1-n,x)$ The first moment about zero is the mean:

$\mu =E[x]=m-{\frac {\kappa -1/\kappa }{\lambda }}$ The variance is:

$\sigma ^{2}=E[x^{2}]-\mu ^{2}={\frac {1+\kappa ^{4}}{\kappa ^{2}\lambda ^{2}}}$ and the skewness is:

${\frac {E[x^{3}]-3\mu \sigma ^{2}-\mu ^{3}}{\sigma ^{3}}}={\frac {2\left(1-\kappa ^{6}\right)}{\left(\kappa ^{4}+1\right)^{3/2}}}$ Generating asymmetric Laplace variates

Asymmetric Laplace variates (X) may be generated from a random variate U drawn from the uniform distribution in the interval (-κ,1/κ) by:

$X=m-{\frac {1}{\lambda \,s\kappa ^{s}}}\log(1-U\,s\kappa ^{S})$ where s=sgn(U).

They may also be generated as the difference of two exponential distributions. If X1 is drawn from exponential distribution with mean and rate (m1,λκ) and X2 is drawn from an exponential distribution with mean and rate (m2,λ/κ) then X1 - X2 is distributed according to the asymmetric Laplace distribution with parameters (m1-m2, λ, κ)

Entropy

The differential entropy of the ALD is

$H=-\int _{-\infty }^{\infty }f_{AL}(x)\log(f_{AL}(x))dx=1-\log \left({\frac {\lambda }{\kappa +1/\kappa }}\right)$ The ALD has the maximum entropy of all distributions with a fixed value (1/λ) of $(x-m)\,s\kappa ^{s}$ where $s=\operatorname {sgn} (x-m)$ .

Alternative Parametrization

An alternative parametrization is made possible by the characteristic function:

$\varphi (t;\mu ,\sigma ,\beta )={\frac {e^{i\mu t}}{1-i\beta \sigma t+\sigma ^{2}t^{2}}}$ where $\mu$ is a location parameter, $\sigma$ is a scale parameter, $\beta$ is an asymmetry parameter. This is specified in Section 2.6.1 and Section 3.1 of Lihn (2015).  Its probability density function is

$f(x;\mu ,\sigma ,\beta )={\frac {1}{2\sigma B_{0}}}{\begin{cases}\exp \left({\frac {x-\mu }{\sigma B^{-}}}\right)&{\text{if }}x<\mu \\[4pt]\exp(-{\frac {x-\mu }{\sigma B^{+}}})&{\text{if }}x\geq \mu \end{cases}}$ where $B_{0}={\sqrt {1+\beta ^{2}/4}}$ and $B^{\pm }=B_{0}\pm \beta /2$ . It follows that $B^{+}B^{-}=1,\P B^{+}-B^{-}=\beta$ .

The n-th moment about $\mu$ is given by

$E[(x-\mu )^{n}]={\frac {\sigma ^{n}n!}{2B_{0}}}((B^{+})^{n+1}+(-1)^{n}(B^{-})^{n+1})$ $E[x]=\mu +\sigma \beta$ The variance is:

$E[x^{2}]-E[x]^{2}=\sigma ^{2}(2+\beta ^{2})$ The skewness is:

${\frac {2\beta (3+\beta ^{2})}{(2+\beta ^{2})^{3/2}}}$ The excess kurtosis is:

${\frac {6(2+4\beta ^{2}+\beta ^{4})}{(2+\beta ^{2})^{2}}}$ For small $\beta$ , the skewness is about $3\beta /{\sqrt {2}}$ . Thus $\beta$ represents skewness in an almost direct way.

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