dmggd: Density of a Multivariate Generalized Gaussian Distribution
In multvardiv: Multivariate Probability Distributions, Statistical Divergence

dmggd

R Documentation

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{\beta}{\Gamma\left(\frac{1}{2 \beta}\right) 2^\frac{1}{2 \beta} \sqrt{\sigma}} \ e^{\displaystyle{-\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. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1080/03610929808832115")}

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)

multvardiv documentation built on April 3, 2025, 6:08 p.m.