dmggd {mggd}

Density of a Multivariate Generalized Gaussian Distribution

Description

Density of the multivariate (p variables) generalized Gaussian distribution (MGGD) with mean vector mu, dispersion matrix Sigma and shape parameter beta.

Usage

dmggd(x, mu, Sigma, beta, tol = 1e-6)

Arguments

x

length p numeric vector.

mu

length p numeric vector. The mean vector.

Sigma

symmetric, positive-definite square matrix of order p. The dispersion matrix.

beta

positive real number. The shape of the distribution.

tol

tolerance (relative to largest variance) for numerical lack of positive-definiteness in Sigma.

Details

The density function of a multivariate generalized Gaussian distribution is given by:

\displaystyle{ f(\mathbf{x}|\boldsymbol{\mu}, \Sigma, \beta) = \frac{\Gamma\left(\frac{p}{2}\right)}{\pi^\frac{p}{2} \Gamma\left(\frac{p}{2 \beta}\right) 2^\frac{p}{2\beta}} \frac{\beta}{|\Sigma|^\frac{1}{2}} e^{-\frac{1}{2}\left((\mathbf{x}-\boldsymbol{\mu})^T \Sigma^{-1} (\mathbf{x}-\boldsymbol{\mu})\right)^\beta} }

When p=1 (univariate case) it becomes:

\displaystyle{ f(x|\mu, \sigma, \beta) = \frac{\Gamma\left(\frac{1}{2}\right)}{\pi^\frac{1}{2} \Gamma\left(\frac{1}{2 \beta}\right) 2^\frac{1}{2\beta}} \frac{\beta}{\sigma^\frac{1}{2}} \ e^{-\frac{1}{2} \left(\frac{(x - \mu)^2}{2 \sigma}\right)^\beta} = \frac{\beta}{\Gamma\left(\frac{1}{2 \beta}\right) 2^\frac{1}{2 \beta} \sqrt{\sigma}} \ e^{-\frac{1}{2} \left(\frac{(x - \mu)^2}{\sigma}\right)^\beta} }

Value

The value of the density.

Author(s)

Pierre Santagostini, Nizar Bouhlel

References

E. Gomez, M. Gomez-Villegas, H. Marin. A Multivariate Generalization of the Power Exponential Family of Distribution. Commun. Statist. 1998, Theory Methods, col. 27, no. 23, p 589-600. doi:10.1080/03610929808832115

See Also

rmggd: random generation from a MGGD.

estparmggd: estimation of the parameters of a MGGD.

plotmggd, contourmggd: plot of the probability density of a bivariate generalised Gaussian distribution.

Examples

mu <- c(0, 1, 4)
Sigma <- matrix(c(0.8, 0.3, 0.2, 0.3, 0.2, 0.1, 0.2, 0.1, 0.2), nrow = 3)
beta <- 0.74
dmggd(c(0, 1, 4), mu, Sigma, beta)
dmggd(c(1, 2, 3), mu, Sigma, beta)