Asymmetric Laplace distribution

A random variable has an asymmetric Laplace(m, λ, κ) distribution if its probability density function is{{cite journal|title=A Multivariate and Asymmetric Generalization of Laplace Distribution|last1=Kozubowski |first1=Tomasz J. |last2=Podgorski |first2=Krzysztof |page=531|year=2000|journal=Computational Statistics|volume=15|issue=4 |doi=10.1007/PL00022717 |s2cid=124839639 |url=https://www.researchgate.net/publication/256065510|access-date=2015-12-29}}{{cite journal |last1=Jammalamadaka |first1=S. Rao |last2=Kozubowski |first2=Tomasz J. |year=2004 |title=New Families of Wrapped Distributions for Modeling Skew Circular Data |journal=Communications in Statistics – Theory and Methods |volume=33 |issue=9 |pages=2059–2074 |url=http://www.pstat.ucsb.edu/faculty/jammalam/html/Some%20Publications/2004_WrappedSkewFamilies_Comm..pdf | doi=10.1081/STA-200026570|s2cid=17024930 |access-date=2011-06-13 }}

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

where s=sgn(x-m), or alternatively:

:$f(x;m,\backslash lambda,\backslash kappa)\; =\; \backslash frac\{\backslash lambda\}\{\backslash kappa+1/\backslash kappa\}$

\begin{cases}

\exp \left( (\lambda/\kappa)(x-m) \right) & \text{if }x < m

\\[4pt]

\exp ( -\lambda\kappa(x-m) ) & \text{if }x \geq m

\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 κ^s simplifies to |x-m| and the distribution simplifies to the Laplace distribution.

The cumulative distribution function is given by:

:$F(x;m,\backslash lambda,\backslash kappa)\; =$

\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}

The ALD characteristic function is given by:

:$\backslash varphi(t;m,\backslash lambda,\backslash kappa)=\backslash frac\{\; e^\{i\; m\; t\}\}\{(1+\backslash frac\{i\; t\; \backslash kappa\}\{\backslash lambda\})\; (1-\backslash frac\{i\; t\}\{\backslash kappa\backslash lambda\})\}$

For m = 0, the ALD is a member of the family of geometric stable distributions with α = 2. It follows that if $\backslash varphi\_1$ and $\backslash varphi\_2$ are two distinct ALD characteristic functions with m = 0, then

:$\backslash varphi=\backslash frac\{1\}\{1/\backslash varphi\_1+1/\backslash varphi\_2-1\}$

is also an ALD characteristic function with location parameter $m=0$. The new scale parameter λ obeys

:$\backslash frac\{1\}\{\backslash lambda^2\}=\backslash frac\{1\}\{\backslash lambda\_1^2\}+\backslash frac\{1\}\{\backslash lambda\_2^2\}$

and the new skewness parameter κ obeys:

:$\backslash frac\{\backslash kappa^2-1\}\{\backslash kappa\backslash lambda\}=\backslash frac\{\backslash kappa\_1^2-1\}\{\backslash kappa\_1\backslash lambda\_1\}+\backslash frac\{\backslash kappa\_2^2-1\}\{\backslash kappa\_2\backslash lambda\_2\}$

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 $E\_n(x)=x^\{n-1\}\backslash Gamma(1-n,x)$

The first moment about zero is the mean:

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

The variance is:

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

and the skewness is:

:$\backslash frac\{E[x^3]-3\backslash mu\backslash sigma^2-\backslash mu^3\}\{\backslash sigma^3\}=\backslash frac\{2\; \backslash left(1-\backslash kappa\; ^6\backslash right)\}\{\backslash left(\backslash kappa\; ^4+1\backslash right)^\{3/2\}\}$

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-\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,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

:$$

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 $(x-m)\backslash ,s\backslash kappa^s$ where $s=\backslash sgn(x-m)$.

An alternative parametrization is made possible by the characteristic function:

$\backslash varphi(t;\backslash mu,\backslash sigma,\backslash beta)=\backslash frac\{\; e^\{i\; \backslash mu\; t\}\}\{1-i\; \backslash beta\; \backslash sigma\; t\; +\backslash sigma^2\; t^2\}$

where $\backslash mu$ is a location parameter, $\backslash sigma$ is a scale parameter, $\backslash beta$ is an asymmetry parameter. This is specified in Section 2.6.1 and Section 3.1 of Lihn (2015).

{{cite journal|last=Lihn|first=Stephen H.-T.|year=2015|title=The Special Elliptic Option Pricing Model and Volatility Smile|url=https://ssrn.com/abstract=2707810|journal=SSRN|page=2707810|access-date=2017-09-05}} Its probability density function is

:$f(x;\backslash mu,\backslash sigma,\backslash beta)\; =\; \backslash frac\{1\}\{2\; \backslash sigma\; B\_0\}$

\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 $B\_0=\backslash sqrt\{1+\backslash beta^2/4\}$

and $B^\backslash pm=B\_0\backslash pm\backslash beta/2$

. It follows that $B^+\; B^-\; =\; 1,\backslash P\; B^+\; -\; B^-\; =\; \backslash beta$

.

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

:$$

E[(x-\mu)^n]=\frac{\sigma^n n!}{2 B_0} ( (B^+)^{n+1} + (-1)^n (B^-)^{n+1} )

The mean about zero is:{{quote|$E[x]=\backslash mu+\backslash sigma\; \backslash beta$

}}The variance is:{{quote|$E[x^2]-E[x]^2=\backslash sigma^2\; (2+\backslash beta^2)$

}}The skewness is:{{quote|$\backslash frac\{2\backslash beta\; (3+\backslash beta^2)\}\{(2+\backslash beta^2)^\{3/2\}\}$

}}The excess kurtosis is:{{quote|$\backslash frac\{6\; (2+4\backslash beta^2+\backslash beta^4)\}\{(2+\backslash beta^2)^\{2\}\}$

}}For small $\backslash beta$, the skewness is about $3\backslash beta/\backslash sqrt\{2\}$ . Thus $\backslash beta$ represents skewness in an almost direct way.

The Asymmetric Laplace distribution is commonly used with an alternative parameterization for performing quantile regression in a Bayesian inference context.{{cite journal|title=Bayesian quantile regression|last1=Yu |first1=Keming |last2=Moyeed |first2=Rana A. |page=437-447|year=2001|journal=Statistics & Probability Letters|volume=54|issue=4 |doi=10.1016/S0167-7152(01)00124-9 |url=https://www.sciencedirect.com/science/article/abs/pii/S0167715201001249|access-date=2022-01-09}} Under this approach, the $\backslash kappa$ parameter describing asymmetry is replaced with a $p$ 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:

:$f(x;m,\backslash lambda,p)\; =\; \backslash frac\{p(1-p)\}\{\backslash lambda\}$

\begin{cases}

\exp \left( -((p-1)/\lambda)(x-m) \right) & \text{if }x \leq m

\\[4pt]

\exp ( -(p/\lambda)(x-m) ) & \text{if }x > 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 ($\backslash mu$) and variance ($\backslash sigma^2$) are calculated as:

:$\backslash mu=m+\backslash frac\{1-2p\}\{p(1-p)\}\backslash lambda$

:$\backslash sigma^2=\backslash frac\{1-2p+2p^2\}\{p^2(1-p)^2\}\backslash lambda^2$

The cumulative distribution function is given{{cite journal|title=A Three-Parameter Asymmetric Laplace Distribution and Its Extension|last1=Yu |first1=Keming |last2=Zhang|first2=Jin |page=1867-1879|year=2005|journal=Communications in Statistics - Theory and Methods|volume=34|issue=9–10 |doi=10.1080/03610920500199018 |s2cid=120827101 |url=https://www.tandfonline.com/doi/abs/10.1080/03610920500199018|access-date=2022-01-30}} by:

:$F(x;m,\backslash lambda,p)\; =$

\begin{cases}

p\exp (\frac{1-p}{\lambda}(x-m)) & \text{if }x \leq m

\\[4pt]

1-(1-p) \exp (\frac{-p}{\lambda}(x-m)) & \text{if }x > 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.{{cite journal|title=A Jump-Diffusion Model for Option Pricing|author=Kou, S.G.|url=https://www.jstor.org/stable/822677|date=August 8, 2002|accessdate=2022-03-01|journal=Management Science|volume=48|issue=8|pages=1086–1101|doi=10.1287/mnsc.48.8.1086.166 |jstor=822677 }} Another example is in neuroscience in which its convolution with normal distribution is considered as a model for brain stopping reaction times.{{cite journal | last1 = Soltanifar | first1 = M| last2 = Escobar | first2 = M |last3 = Dupuis | first3 = A|last4 = Chevrier | first4 = A|last5 = Schachar | first5 = R |year = 2022 | title = The Asymmetric Laplace Gaussian (ALG) Distribution as the Descriptive Model for the Internal Proactive Inhibition in the Standard Stop Signal Task | doi = 10.3390/brainsci12060730 | pmid = 35741615| journal = Brain Sciences | volume = 12 | issue = 6| page = 730| pmc = 9221528| doi-access = free }}

{{Reflist}}

{{ProbDistributions|continuous-infinite}}

{{DEFAULTSORT:Asymmetric Laplace Distribution}}

Category:Continuous distributions

Category:Exponential family distributions

Category:Location-scale family probability distributions

Category:Geometric stable distributions