Asymmetric Laplace distribution

Asymmetric Laplace
	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.
Parameters	location (real); scale (real); asymmetry (real)
Support
PDF	(see article)
CDF	(see article)
Mean
Median
Variance
Skewness
Entropy
CF

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.

Characterization

Probability density function

A random variable has an asymmetric Laplace(m, λ, κ) distribution if its probability density function is^[1]^[2]

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<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.

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(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). ^[3] 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})

The mean about zero is:

$E[x]=\mu +\sigma \beta$

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.

References

↑ Kozubowski, Tomasz J.; Podgorski, Krzysztof (2000). "A Multivariate and Asymmetric Generalization of Laplace Distribution" (PDF). Computational Statistics. 15: 531. Retrieved 2015-12-29.
↑ Jammalamadaka, S. Rao; Kozubowski, Tomasz J. (2004). "New Families of Wrapped Distributions for Modeling Skew Circular Data" (PDF). Communications in Statistics – Theory and Methods. 33 (9): 2059–2074. doi:10.1081/STA-200026570. Retrieved 2011-06-13.
↑ Lihn, Stephen H.-T. (2015). "The Special Elliptic Option Pricing Model and Volatility Smile". SSRN: 2707810. Retrieved 2017-09-05.

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[KOZ2000-1] Kozubowski, Tomasz J.; Podgorski, Krzysztof (2000). "A Multivariate and Asymmetric Generalization of Laplace Distribution" (PDF). Computational Statistics. 15: 531. Retrieved 2015-12-29.

[JAM-2] Jammalamadaka, S. Rao; Kozubowski, Tomasz J. (2004). "New Families of Wrapped Distributions for Modeling Skew Circular Data" (PDF). Communications in Statistics – Theory and Methods. 33 (9): 2059–2074. doi:10.1081/STA-200026570. Retrieved 2011-06-13.

[LIHN2015-3] Lihn, Stephen H.-T. (2015). "The Special Elliptic Option Pricing Model and Volatility Smile". SSRN: 2707810. Retrieved 2017-09-05.

Probability distributions
List
Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot
Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta
Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle
Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda
Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt
Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Marchenko–Pastur q-exponential q-Gaussian q-Weibull shifted log-logistic Tukey lambda
Mixed continuous-discrete univariate	rectified Gaussian
Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate Laplace multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart
Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham
Degenerate and singular	Degenerate Dirac delta function Singular Cantor
Families	Circular compound Poisson elliptical exponential natural exponential location–scale maximum entropy mixture Pearson Tweedie wrapped

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.
Parameters	$m$ location (real) $\lambda >0$ scale (real) $\kappa > 0$ asymmetry (real)
Support	$x \in (-\infty; +\infty)\,$
PDF	(see article)
CDF	(see article)
Mean	$m+{\frac {1-\kappa ^{2}}{\lambda \kappa }}$
Median	$m+(\kappa /\lambda )\log \left({\frac {1+\kappa ^{2}}{2\kappa ^{2}}}\right)$
Variance	${\frac {1+\kappa ^{4}}{\lambda ^{2}\kappa ^{2}}}$
Skewness	${\frac {2\left(1-\kappa ^{6}\right)}{\left(\kappa ^{4}+1\right)^{3/2}}}$
Entropy	$\log \left(e\,{\frac {1+\kappa ^{2}}{\kappa \lambda }}\right)$
CF	${\frac {e^{imt}}{(1+{\frac {it\kappa }{\lambda }})(1-{\frac {it}{\kappa \lambda }})}}$