# 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 |