In the mathematical theory of probability,multivariate Laplace distributions are extensions of theLaplace distribution and theasymmetric Laplace distribution to multiple variables. Themarginal distributions of symmetric multivariate Laplace distribution variables are Laplace distributions. The marginal distributions of asymmetric multivariate Laplace distribution variables are asymmetric Laplace distributions.^[1]

Multivariate Laplace (symmetric)
Parameters	μ ∈Rk —location Σ ∈Rk×k —covariance (positive-definite matrix)
Support	x ∈μ + span(Σ) ⊆Rk
PDF	If${\displaystyle \mathit{\mu }=\mathbf{0}}$, ${\displaystyle \frac{2}{(2\pi {)}^{k/2}{\left|\mathbf{\Sigma }\right|}^{1/2}}{\left(\frac{{\mathbf{x}}^{\prime }{\mathbf{\Sigma }}^{-1}\mathbf{x}}{2}\right)}^{v/2}{K}_v\left(\sqrt{2{\mathbf{x}}^{\prime }{\mathbf{\Sigma }}^{-1}\mathbf{x}}\right),}$ where${\displaystyle v=(2-k)/2}$ and${\displaystyle K_v}$ is themodified Bessel function of the second kind.
Mean	μ
Mode	μ
Variance	Σ
Skewness	0
CF	${\displaystyle \frac{\exp (i{\mathit{\mu }}^{\prime }\mathbf{t})}{1+\frac{1}{2}{\mathbf{t}}^{\prime }\mathbf{\Sigma }\mathbf{t}}}$

A typical characterization of the symmetric multivariate Laplace distribution has thecharacteristic function:

${\displaystyle \phi (t;\mathit{\mu },\mathbf{\Sigma })=\frac{\exp (i{\mathit{\mu }}^{\prime }\mathbf{t})}{1+\frac{1}{2}{\mathbf{t}}^{\prime }\mathbf{\Sigma }\mathbf{t}},}$

where${\displaystyle \mathit{\mu }}$ is the vector ofmeans for each variable and${\displaystyle \mathbf{\Sigma }}$ is thecovariance matrix.^[2]

Unlike themultivariate normal distribution, even if the covariance matrix has zerocovariance andcorrelation the variables are not independent.^[1] The symmetric multivariate Laplace distribution iselliptical.^[1]

If${\displaystyle \mathit{\mu }=\mathbf{0}}$, theprobability density function (pdf) for ak-dimensional multivariate Laplace distribution becomes:

${\displaystyle f_{\mathbf{x}}(x_1,\dots ,x_k)=\frac{2}{(2\pi {)}^{k/2}|\mathbf{\Sigma }{|}^{0.5}}{\left(\frac{{\mathbf{x}}^{\prime }{\mathbf{\Sigma }}^{-1}\mathbf{x}}{2}\right)}^{v/2}{K}_v\left(\sqrt{2{\mathbf{x}}^{\prime }{\mathbf{\Sigma }}^{-1}\mathbf{x}}\right),}$

where:

${\displaystyle v=(2-k)/2}$ and${\displaystyle K_v}$ is themodified Bessel function of the second kind.^[1]

In the correlated bivariate case, i.e.,k = 2, with${\displaystyle {\mu }_1={\mu }_2=0}$ the pdf reduces to:

${\displaystyle f_{\mathbf{x}}(x_1,x_2)=\frac{1}{\pi {\sigma }_1{\sigma }_2\sqrt{1-{\rho }^2}}{K}_0\left(\sqrt{\frac{2\left(\frac{x_1^2}{{\sigma }_1^2}-\frac{2\rho x_1{x}_2}{{\sigma }_1{\sigma }_2}+\frac{x_2^2}{{\sigma }_2^2}\right)}{1-{\rho }^2}}\right),}$

where:

${\displaystyle {\sigma }_1}$ and${\displaystyle {\sigma }_2}$ are thestandard deviations of${\displaystyle x_1}$ and${\displaystyle x_2}$, respectively, and${\displaystyle \rho }$ is thecorrelation coefficient of${\displaystyle x_1}$ and${\displaystyle x_2}$.^[1]

For the uncorrelated bivariate Laplace case, that isk = 2,${\displaystyle {\mu }_1={\mu }_2=\rho =0}$ and${\displaystyle {\sigma }_1={\sigma }_2=1}$, the pdf becomes:

${\displaystyle f_{\mathbf{x}}(x_1,x_2)=\frac{1}{\pi }{K}_0\left(\sqrt{2(x_1^2+x_2^2)}\right).}$^[1]

Multivariate Laplace (asymmetric)
Parameters	μ ∈Rk —location Σ ∈Rk×k —covariance (positive-definite matrix)
Support	x ∈μ + span(Σ) ⊆Rk
PDF	${\displaystyle \frac{2e^{{\mathbf{x}}^{\prime }{\mathbf{\Sigma }}^{-1}\mathit{\mu }}}{(2\pi {)}^{\frac{k}{2}}|\mathbf{\Sigma }{|}^{0.5}}(\frac{{\mathbf{x}}^{\prime }{\mathbf{\Sigma }}^{-1}\mathbf{x}}{2+{\mathit{\mu }}^{\prime }{\mathbf{\Sigma }}^{-1}\mathit{\mu }}{)}^{\frac{v}{2}}{K}_v(\sqrt{(2+{\mathit{\mu }}^{\prime }{\mathbf{\Sigma }}^{-1}\mathit{\mu })({\mathbf{x}}^{\prime }{\mathbf{\Sigma }}^{-1}\mathbf{x})})}$ where${\displaystyle v=(2-k)/2}$ and${\displaystyle K_v}$ is themodified Bessel function of the second kind.
Mean	μ
Variance	Σ +μ 'μ
Skewness	non-zero unlessμ=0
CF	${\displaystyle \frac{1}{1+\frac{1}{2}{\mathbf{t}}^{\prime }\mathbf{\Sigma }\mathbf{t}-i\mathit{\mu }\mathbf{t}}}$

A typical characterization of the asymmetric multivariate Laplace distribution has thecharacteristic function:

${\displaystyle \phi (t;\mathit{\mu },\mathbf{\Sigma })=\frac{1}{1+\frac{1}{2}{\mathbf{t}}^{\prime }\mathbf{\Sigma }\mathbf{t}-i\mathit{\mu }\mathbf{t}}.}$^[1]

As with the symmetric multivariate Laplace distribution, the asymmetric multivariate Laplace distribution has mean${\displaystyle \mathit{\mu }}$, but the covariance becomes${\displaystyle \mathbf{\Sigma }+{\mathit{\mu }}^{\prime }\mathit{\mu }}$.^[3] The asymmetric multivariate Laplace distribution is not elliptical unless${\displaystyle \mathit{\mu }=\mathbf{0}}$, in which case the distribution reduces to the symmetric multivariate Laplace distribution with${\displaystyle \mathit{\mu }=\mathbf{0}}$.^[1]

Theprobability density function (pdf) for ak-dimensional asymmetric multivariate Laplace distribution is:

${\displaystyle f_{\mathbf{x}}(x_1,\dots ,x_k)=\frac{2e^{{\mathbf{x}}^{\prime }{\mathbf{\Sigma }}^{-1}\mathit{\mu }}}{(2\pi {)}^{k/2}|\mathbf{\Sigma }{|}^{0.5}}(\frac{{\mathbf{x}}^{\prime }{\mathbf{\Sigma }}^{-1}\mathbf{x}}{2+{\mathit{\mu }}^{\prime }{\mathbf{\Sigma }}^{-1}\mathit{\mu }}{)}^{v/2}{K}_v(\sqrt{(2+{\mathit{\mu }}^{\prime }{\mathbf{\Sigma }}^{-1}\mathit{\mu })({\mathbf{x}}^{\prime }{\mathbf{\Sigma }}^{-1}\mathbf{x})}),}$

where:

${\displaystyle v=(2-k)/2}$ and${\displaystyle K_v}$ is themodified Bessel function of the second kind.^[1]

The asymmetric Laplace distribution, including the special case of${\displaystyle \mathit{\mu }=\mathbf{0}}$, is an example of ageometric stable distribution.^[3] It represents the limiting distribution for a sum ofindependent, identically distributed random variables with finite variance and covariance where the number of elements to be summed is itself an independent random variable distributed according to ageometric distribution.^[1] Such geometric sums can arise in practical applications within biology, economics and insurance.^[1] The distribution may also be applicable in broader situations to model multivariate data with heavier tails than a normal distribution but finitemoments.^[1]

The relationship between theexponential distribution and theLaplace distribution allows for a simple method for simulating bivariate asymmetric Laplace variables (including for the case of${\displaystyle \mathit{\mu }=\mathbf{0}}$). Simulate a bivariate normal random variable vector${\displaystyle \mathbf{Y}}$ from a distribution with${\displaystyle {\mu }_1={\mu }_2=0}$ and covariance matrix${\displaystyle \mathbf{\Sigma }}$. Independently simulate an exponential random variable${\displaystyle \mathbf{W}}$ from an Exp(1) distribution.${\displaystyle \mathbf{X}=\sqrt{W}\mathbf{Y}+W\mathit{\mu }}$ will be distributed (asymmetric) bivariate Laplace with mean${\displaystyle \mathit{\mu }}$ and covariance matrix${\displaystyle \mathbf{\Sigma }}$.^[1]

• Probability distributions
• Multivariate continuous distributions
• Geometric stable distributions

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