Multivariate Laplace distribution

Multivariate Laplace (symmetric)

Parameters	μ ∈ Rk — location Σ ∈ Rk×k — covariance (positive-definite matrix)
Support	x ∈ μ + span(Σ) ⊆ Rk
PDF	If ${\displaystyle {\boldsymbol {\mu }}=\mathbf {0} }$, ${\displaystyle {\frac {2}{(2\pi )^{k/2}\left|{\boldsymbol {\Sigma }}\right|^{1/2}}}\left({\frac {\mathbf {x} '{\boldsymbol {\Sigma }}^{-1}\mathbf {x} }{2}}\right)^{v/2}K_{v}\left({\sqrt {2\mathbf {x} '{\boldsymbol {\Sigma }}^{-1}\mathbf {x} }}\right),}$ where ${\displaystyle v=(2-k)/2}$ and ${\displaystyle K_{v}}$ is the modified Bessel function of the second kind.
Mean	μ
Mode	μ
Variance	Σ
Skewness	0
CF	${\displaystyle {\frac {\exp(i{\boldsymbol {\mu }}'\mathbf {t} )}{1+{\tfrac {1}{2}}\mathbf {t} '{\boldsymbol {\Sigma }}\mathbf {t} }}}$

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

${\displaystyle \varphi (t;{\boldsymbol {\mu }},{\boldsymbol {\Sigma }})={\frac {\exp(i{\boldsymbol {\mu }}'\mathbf {t} )}{1+{\tfrac {1}{2}}\mathbf {t} '{\boldsymbol {\Sigma }}\mathbf {t} }},}$

where ${\displaystyle {\boldsymbol {\mu }}}$ is the vector of means for each variable and ${\displaystyle {\boldsymbol {\Sigma }}}$ is the covariance matrix.^[2]

Unlike the multivariate normal distribution, even if the covariance matrix has zero covariance and correlation the variables are not independent.^[1] The symmetric multivariate Laplace distribution is elliptical.^[1]

If ${\displaystyle {\boldsymbol {\mu }}=\mathbf {0} }$, the probability density function (pdf) for a k-dimensional multivariate Laplace distribution becomes:

${\displaystyle f_{\mathbf {x} }(x_{1},\ldots ,x_{k})={\frac {2}{(2\pi )^{k/2}|{\boldsymbol {\Sigma }}|^{0.5}}}\left({\frac {\mathbf {x} '{\boldsymbol {\Sigma }}^{-1}\mathbf {x} }{2}}\right)^{v/2}K_{v}\left({\sqrt {2\mathbf {x} '{\boldsymbol {\Sigma }}^{-1}\mathbf {x} }}\right),}$

where:

${\displaystyle v=(2-k)/2}$ and ${\displaystyle K_{v}}$ is the modified 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 the standard deviations of ${\displaystyle x_{1}}$ and ${\displaystyle x_{2}}$, respectively, and ${\displaystyle \rho }$ is the correlation coefficient of ${\displaystyle x_{1}}$ and ${\displaystyle x_{2}}$.^[1]

For the uncorrelated bivariate Laplace case, that is k = 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	${\displaystyle {\frac {2e^{\mathbf {x} '{\boldsymbol {\Sigma }}^{-1}{\boldsymbol {\mu }}}}{(2\pi )^{\frac {k}{2}}|{\boldsymbol {\Sigma }}|^{0.5}}}{\Big (}{\frac {\mathbf {x} '{\boldsymbol {\Sigma }}^{-1}\mathbf {x} }{2+{\boldsymbol {\mu }}'{\boldsymbol {\Sigma }}^{-1}{\boldsymbol {\mu }}}}{\Big )}^{\frac {v}{2}}K_{v}{\Big (}{\sqrt {(2+{\boldsymbol {\mu }}'{\boldsymbol {\Sigma }}^{-1}{\boldsymbol {\mu }})(\mathbf {x} '{\boldsymbol {\Sigma }}^{-1}\mathbf {x} )}}{\Big )}}$ where ${\displaystyle v=(2-k)/2}$ and ${\displaystyle K_{v}}$ is the modified Bessel function of the second kind.
Mean	μ
Variance	Σ + μ ' μ
Skewness	non-zero unless μ=0
CF	${\displaystyle {\frac {1}{1+{\tfrac {1}{2}}\mathbf {t} '{\boldsymbol {\Sigma }}\mathbf {t} -i{\boldsymbol {\mu }}\mathbf {t} }}}$

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

${\displaystyle \varphi (t;{\boldsymbol {\mu }},{\boldsymbol {\Sigma }})={\frac {1}{1+{\tfrac {1}{2}}\mathbf {t} '{\boldsymbol {\Sigma }}\mathbf {t} -i{\boldsymbol {\mu }}\mathbf {t} }}.}$^[1]

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

The probability density function (pdf) for a k-dimensional asymmetric multivariate Laplace distribution is:

${\displaystyle f_{\mathbf {x} }(x_{1},\ldots ,x_{k})={\frac {2e^{\mathbf {x} '{\boldsymbol {\Sigma }}^{-1}{\boldsymbol {\mu }}}}{(2\pi )^{k/2}|{\boldsymbol {\Sigma }}|^{0.5}}}{\Big (}{\frac {\mathbf {x} '{\boldsymbol {\Sigma }}^{-1}\mathbf {x} }{2+{\boldsymbol {\mu }}'{\boldsymbol {\Sigma }}^{-1}{\boldsymbol {\mu }}}}{\Big )}^{v/2}K_{v}{\Big (}{\sqrt {(2+{\boldsymbol {\mu }}'{\boldsymbol {\Sigma }}^{-1}{\boldsymbol {\mu }})(\mathbf {x} '{\boldsymbol {\Sigma }}^{-1}\mathbf {x} )}}{\Big )},}$

where:

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

The asymmetric Laplace distribution, including the special case of ${\displaystyle {\boldsymbol {\mu }}=\mathbf {0} }$, is an example of a geometric stable distribution.^[3] It represents the limiting distribution for a sum of independent, 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 a geometric 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 finite moments.^[1]

The relationship between the exponential distribution and the Laplace distribution allows for a simple method for simulating bivariate asymmetric Laplace variables (including for the case of ${\displaystyle {\boldsymbol {\mu }}=\mathbf {0} }$). Simulate a bivariate normal random variable vector ${\displaystyle \mathbf {Y} }$ from a distribution with ${\displaystyle \mu _{1}=\mu _{2}=0}$ and covariance matrix ${\displaystyle {\boldsymbol {\Sigma }}}$. Independently simulate an exponential random variable ${\displaystyle \mathbf {W} }$ from an Exp(1) distribution. ${\displaystyle \mathbf {X} ={\sqrt {W}}\mathbf {Y} +W{\boldsymbol {\mu }}}$ will be distributed (asymmetric) bivariate Laplace with mean ${\displaystyle {\boldsymbol {\mu }}}$ and covariance matrix ${\displaystyle {\boldsymbol {\Sigma }}}$.^[1]