Wishart distribution
Definition taught in Estimation and inference for the multivariate normal, part of Statistical learning
Wishart distribution
| Type | Definition |
| Taught in | Estimation and inference for the multivariate normal |
| Part | Statistical learning |
Let \mathbf{U}_1,\ldots,\mathbf{U}_\nu be independent \operatorname{N}_p(\mathbf{0},\Sigma) vectors, with \nu\ge p and \Sigma positive definite. The matrix
W=\sum_{m=1}^\nu\mathbf{U}_m\mathbf{U}_m^\top
has the Wishart distribution W\sim W_p(\Sigma,\nu), with density
w(W;\Sigma,\nu) =\frac{|W|^{(\nu-p-1)/2}\exp\bigl\{-\tfrac12\operatorname{tr}(W\Sigma^{-1})\bigr\}} {2^{\nu p/2}\,\pi^{p(p-1)/4}\,|\Sigma|^{\nu/2}\prod_{j=1}^p\Gamma\bigl((\nu+1-j)/2\bigr)}
for positive definite W, and 0 otherwise.
Used in
- Taught in Estimation and inference for the multivariate normal, where the derivation and examples are.
- All 9 questions on Estimation and inference for the multivariate normal
See also
Positive definite matrix · Two-sample Hotelling test · Invariance of the maximum likelihood estimator · Sampling law of the normal mean and scatter · Multivariate central limit theorem · The limit of Hotelling’s statistic · Null distribution of Hotelling’s · Likelihood of a multivariate normal sample