multivariate Laplace distribution

{{Probability distribution

| name = Multivariate Laplace (symmetric)

| type = multivariate

| pdf_image =

| cdf_image =

| notation =

| parameters = μ ∈ R^k — location
Σ ∈ R^k×k — covariance (positive-definite matrix)

| support = x ∈ μ + span(Σ) ⊆ R^k

| pdf = :If $\backslash boldsymbol\backslash mu\; =\; \backslash mathbf\{0\}$,
$\backslash frac\; 2\; \{(2\backslash pi)^\{k/2\}\; \backslash left|\backslash boldsymbol\backslash Sigma\backslash right|^\{1/2\}\}\; \backslash left(\; \backslash frac\; \{\backslash mathbf\{x\}\text{'}\backslash boldsymbol\backslash Sigma^\{-1\}\; \backslash mathbf\{x\}\}\{2\}\; \backslash right)^\{v/2\}\; K\_v\; \backslash left(\backslash sqrt\{2\backslash mathbf\{x\}\text{'}\backslash boldsymbol\backslash Sigma^\{-1\}\; \backslash mathbf\{x\}\}\; \backslash right),$
where $v\; =\; (2\; -\; k)\; /\; 2$ and $K\_v$ is the modified Bessel function of the second kind.

| mean = μ

| median =

| mode = μ

| variance = Σ

| skewness =0

| kurtosis =

| entropy =

| mgf =

| char = $\backslash frac\{\backslash exp\; (\; i\backslash boldsymbol\backslash mu\text{'}\backslash mathbf\{t\}\; )\}\{1\; +\; \backslash tfrac\{1\}\{2\}\; \backslash mathbf\{t\}\text{'}\backslash boldsymbol\backslash Sigma\; \backslash mathbf\{t\}\; \}$

}}

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

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

where $\backslash boldsymbol\backslash mu$ is the vector of means for each variable and $\backslash boldsymbol\backslash Sigma$ is the covariance matrix.{{cite journal|title=Goodness-of-fit tests for multivariate Laplace distributions|author=Fragiadakis, Konstantinos & Meintanis, Simos G.|journal=Mathematical and Computer Modelling|volume=53|issue=5–6|date=March 2011|pages=769–779|doi=10.1016/j.mcm.2010.10.014|doi-access=free}}

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

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

:$f\_\{\backslash mathbf\; x\}(x\_1,\backslash ldots,x\_k)\; =\; \backslash frac\; 2\; \{(2\backslash pi)^\{k/2\}\; |\backslash boldsymbol\backslash Sigma|^\{0.5\}\}\; \backslash left(\; \backslash frac\; \{\backslash mathbf\{x\}\text{'}\backslash boldsymbol\backslash Sigma^\{-1\}\; \backslash mathbf\{x\}\}\{2\}\; \backslash right)^\{v/2\}\; K\_v\; \backslash left(\backslash sqrt\{2\backslash mathbf\{x\}\text{'}\backslash boldsymbol\backslash Sigma^\{-1\}\; \backslash mathbf\{x\}\}\; \backslash right),$

where:

$v\; =\; (2\; -\; k)\; /\; 2$ and $K\_v$ is the modified Bessel function of the second kind.

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

:$f\_\{\backslash mathbf\; x\}(x\_1,x\_2)\; =\; \backslash frac\; 1\; \{\backslash pi\; \backslash sigma\_1\; \backslash sigma\_2\; \backslash sqrt\{1-\backslash rho^2\}\}\; K\_0\; \backslash left(\; \backslash sqrt\; \{\; \backslash frac\; \{\; 2\; \backslash left(\; \backslash frac\; \{x\_1^2\}\{\backslash sigma\_1^2\}\; -\; \backslash frac\{2\; \backslash rho\; x\_1\; x\_2\}\{\backslash sigma\_1\; \backslash sigma\_2\}\; +\; \backslash frac\; \{x\_2^2\}\{\backslash sigma\_2^2\}\; \backslash right)\}\; \{\; 1-\backslash rho^2\; \}\; \}\; \backslash right),$

where:

$\backslash sigma\_1$ and $\backslash sigma\_2$ are the standard deviations of $x\_1$ and $x\_2$, respectively, and $\backslash rho$ is the correlation coefficient of $x\_1$ and $x\_2$.

For the uncorrelated bivariate Laplace case, that is k = 2, $\backslash mu\_1\; =\; \backslash mu\_2\; =\; \backslash rho\; =\; 0$ and $\backslash sigma\_1\; =\; \backslash sigma\_2\; =\; 1$, the pdf becomes:

:$f\_\{\backslash mathbf\; x\}(x\_1,x\_2)\; =\; \backslash frac\; 1\; \backslash pi\; K\_0\; \backslash left(\; \backslash sqrt\; \{\; 2(x\_1^2\; +\; x\_2^2)\; \}\; \backslash right).$

{{Probability distribution

| name = Multivariate Laplace (asymmetric)

| type = multivariate

| pdf_image =

| cdf_image =

| notation =

| parameters = μ ∈ R^k — location
Σ ∈ R^k×k — covariance (positive-definite matrix)

| support = x ∈ μ + span(Σ) ⊆ R^k

| pdf = $\backslash frac\; \{2\; e^\{\backslash mathbf\{x\}\text{'}\backslash boldsymbol\backslash Sigma^\{-1\}\; \backslash boldsymbol\backslash mu\}\; \}\{(2\backslash pi)^\{\backslash frac\{k\}\{2\}\}\; |\backslash boldsymbol\backslash Sigma|^\{0.5\}\}\; \backslash Big(\; \backslash frac\; \{\backslash mathbf\{x\}\text{'}\backslash boldsymbol\backslash Sigma^\{-1\}\; \backslash mathbf\{x\}\}\{2\; +\; \backslash boldsymbol\backslash mu\text{'}\backslash boldsymbol\backslash Sigma^\{-1\}\; \backslash boldsymbol\backslash mu\}\; \backslash Big)^\backslash frac\{v\}\{2\}\; K\_v\; \backslash Big(\backslash sqrt\{(2\; +\; \backslash boldsymbol\backslash mu\text{'}\backslash boldsymbol\backslash Sigma^\{-1\}\; \backslash boldsymbol\backslash mu)(\backslash mathbf\{x\}\text{'}\backslash boldsymbol\backslash Sigma^\{-1\}\; \backslash mathbf\{x\})\}\; \backslash Big)$
where $v\; =\; (2\; -\; k)\; /\; 2$ and $K\_v$ is the modified Bessel function of the second kind.

| mean = μ

| median =

| mode =

| variance = Σ + μ ' μ

| skewness = non-zero unless μ=0

| kurtosis =

| entropy =

| mgf =

| char = $\backslash frac\{1\}\{1\; +\; \backslash tfrac\{1\}\{2\}\; \backslash mathbf\{t\}\text{'}\backslash boldsymbol\backslash Sigma\; \backslash mathbf\{t\}\; -\; i\backslash boldsymbol\backslash mu\backslash mathbf\{t\}\; \}$

}}

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

: $\backslash varphi(t;\backslash boldsymbol\backslash mu,\backslash boldsymbol\backslash Sigma)\; =\; \backslash frac\{1\}\{1\; +\; \backslash tfrac\{1\}\{2\}\; \backslash mathbf\{t\}\text{'}\backslash boldsymbol\backslash Sigma\; \backslash mathbf\{t\}\; -\; i\backslash boldsymbol\backslash mu\backslash mathbf\{t\}\; \}.$

As with the symmetric multivariate Laplace distribution, the asymmetric multivariate Laplace distribution has mean $\backslash boldsymbol\backslash mu$, but the covariance becomes $\backslash boldsymbol\backslash Sigma\; +\; \backslash boldsymbol\backslash mu\text{'}\backslash boldsymbol\backslash mu$.{{cite journal|title=Multivariate Generalize Laplace Distributions and Related Random Fields|author1=Kozubowski, Tomasz J. |author2=Podgorski, Krzysztof |author3=Rychlik, Igor |journal=Journal of Multivariate Analysis |publisher=University of Gothenburg|year=2010|volume=113 |pages=59–72 |doi=10.1016/j.jmva.2012.02.010 |s2cid=206252976 |doi-access=free }} The asymmetric multivariate Laplace distribution is not elliptical unless $\backslash boldsymbol\backslash mu\; =\; \backslash mathbf\{0\}$, in which case the distribution reduces to the symmetric multivariate Laplace distribution with $\backslash boldsymbol\backslash mu\; =\; \backslash mathbf\{0\}$.

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

:$f\_\{\backslash mathbf\; x\}(x\_1,\backslash ldots,x\_k)\; =\; \backslash frac\; \{2\; e^\{\backslash mathbf\{x\}\text{'}\backslash boldsymbol\backslash Sigma^\{-1\}\; \backslash boldsymbol\backslash mu\}\; \}\{(2\backslash pi)^\{k/2\}\; |\backslash boldsymbol\backslash Sigma|^\{0.5\}\}\; \backslash Big(\; \backslash frac\; \{\backslash mathbf\{x\}\text{'}\backslash boldsymbol\backslash Sigma^\{-1\}\; \backslash mathbf\{x\}\}\{2\; +\; \backslash boldsymbol\backslash mu\text{'}\backslash boldsymbol\backslash Sigma^\{-1\}\; \backslash boldsymbol\backslash mu\}\; \backslash Big)^\{v/2\}\; K\_v\; \backslash Big(\backslash sqrt\{(2\; +\; \backslash boldsymbol\backslash mu\text{'}\backslash boldsymbol\backslash Sigma^\{-1\}\; \backslash boldsymbol\backslash mu)(\backslash mathbf\{x\}\text{'}\backslash boldsymbol\backslash Sigma^\{-1\}\; \backslash mathbf\{x\})\}\; \backslash Big),$

where:

$v\; =\; (2\; -\; k)\; /\; 2$ and $K\_v$ is the modified Bessel function of the second kind.

The asymmetric Laplace distribution, including the special case of $\backslash boldsymbol\backslash mu\; =\; \backslash mathbf\{0\}$, is an example of a geometric stable distribution. 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. Such geometric sums can arise in practical applications within biology, economics and insurance. The distribution may also be applicable in broader situations to model multivariate data with heavier tails than a normal distribution but finite moments.

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 $\backslash boldsymbol\backslash mu\; =\; \backslash mathbf\{0\}$). Simulate a bivariate normal random variable vector $\backslash mathbf\{Y\}$ from a distribution with $\backslash mu\_1=\backslash mu\_2=0$ and covariance matrix $\backslash boldsymbol\backslash Sigma$. Independently simulate an exponential random variable $\backslash mathbf\{W\}$ from an Exp(1) distribution. $\backslash mathbf\{X\}\; =\; \backslash sqrt\{W\}\; \backslash mathbf\{Y\}\; +\; W\; \backslash boldsymbol\backslash mu$ will be distributed (asymmetric) bivariate Laplace with mean $\backslash boldsymbol\backslash mu$ and covariance matrix $\backslash boldsymbol\backslash Sigma$.

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{{ProbDistributions|multivariate|state=collapsed}}

Category:Probability distributions

Category:Multivariate continuous distributions

Category:Geometric stable distributions