Wishart distribution

Let $U_i\sim N_p({\mu }_i,\mathrm{\Sigma }),i=1,\mathrm{\dots },k$ be independent $p$-dimensional random variables, which are multivariate normally distributed (http://planetmath.org/jointnormaldistribution). Let $S={\sum }_{i=1}^k{U}_{i}U_i{}^T$. Let $M$ be the $k\times p$ matrix with ${\mu }_1,\mathrm{\dots },{\mu }_k$ as rows. Then the joint distribution of the elements of $S$ is said to be a Wishart distribution on $k$ of freedom , and is denoted by $W_p(k,\mathrm{\Sigma },M)$. If $M=0$, the distribution is said to be central and is denoted by $W_p(k,\mathrm{\Sigma })$. The Wishart distribution is a multivariate generalization of the ${\chi }^2$ distribution.

$W_p$ has a density function when $k\ge p$.

Title	Wishart distribution
Canonical name	WishartDistribution
Date of creation	2013-03-22 16:12:31
Last modified on	2013-03-22 16:12:31
Owner	Mathprof (13753)
Last modified by	Mathprof (13753)
Numerical id	9
Author	Mathprof (13753)
Entry type	Definition
Classification	msc 62H05
Defines	central Wishart distribution