Null distribution of Hotelling’s
Theorem taught in Estimation and inference for the multivariate normal, part of Statistical learning
Null distribution of Hotelling’s
| Type | Theorem |
| Taught in | Estimation and inference for the multivariate normal |
| Part | Statistical learning |
For n>p independent observations from \operatorname{N}_p(\boldsymbol{\mu},\Sigma), the statistic
T^2=n(\overline{\mathbf{X}}-\boldsymbol{\mu}_0)^\top S^{-1}(\overline{\mathbf{X}}-\boldsymbol{\mu}_0), \qquad S=\frac{1}{n-1}A,
satisfies
F=\frac{n-p}{p(n-1)}\,T^2\sim F_{p,\,n-p}
when \boldsymbol{\mu}=\boldsymbol{\mu}_0. Reject H_0:\boldsymbol{\mu}=\boldsymbol{\mu}_0 at level \alpha when F>F_{\alpha;\,p,\,n-p}.
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
Statistic · Invariance of the maximum likelihood estimator · Wishart distribution · Sampling law of the normal mean and scatter · Multivariate central limit theorem · The limit of Hotelling’s statistic · Two-sample Hotelling test · Likelihood of a multivariate normal sample