Details: density when p=1

R/mvdggd.R

@@ -16,7 +16,10 @@ mvdggd <- function(x, mu, Sigma, beta, tol = 1e-6) {
   #' 
   #' @details The density function of a multivariate generalized Gaussian distribution is given by:
   #' \deqn{ \displaystyle{ f(x|\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((x-\mu)^T \Sigma^{-1} (x-\mu)\right)^\beta} } }
-  #'
+  #' 
+  #' When \eqn{p=1} (univariate case) it becomes:
+  #' \deqn{ \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^{-\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^{-\left(\frac{(x - \mu)^2}{2 \sigma}\right)^\beta} } }
+  #' 
   #' @author 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.

man/mvdggd.Rd

@@ -27,6 +27,9 @@ with mean vector \code{mu}, dispersion matrix \code{Sigma} and shape parameter \
 \details{
 The density function of a multivariate generalized Gaussian distribution is given by:
 \deqn{ \displaystyle{ f(x|\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((x-\mu)^T \Sigma^{-1} (x-\mu)\right)^\beta} } }
+
+When \eqn{p=1} (univariate case) it becomes:
+\deqn{ \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^{-\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^{-\left(\frac{(x - \mu)^2}{2 \sigma}\right)^\beta} } }
 }
 \examples{
 mu <- c(0, 1, 4)