The limit of Hotelling’s statistic
Remark taught in Estimation and inference for the multivariate normal, part of Statistical learning
The limit of Hotelling’s statistic
| Type | Remark |
| Taught in | Estimation and inference for the multivariate normal |
| Part | Statistical learning |
Under the null \boldsymbol{\mu}=\boldsymbol{\mu}_0, for fixed p and positive definite \Sigma, if S is consistent for \Sigma then
n(\overline{\mathbf{X}}-\boldsymbol{\mu}_0)^\top S^{-1}(\overline{\mathbf{X}}-\boldsymbol{\mu}_0)
is asymptotically \chi^2_p. The multivariate central limit theorem makes \sqrt{n}\,\Sigma^{-1/2}(\overline{\mathbf{X}}-\boldsymbol{\mu}_0) asymptotically \operatorname{N}_p(\mathbf{0},I), and S\to\Sigma in probability replaces \Sigma by S in the quadratic form.
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 · Multivariate central limit theorem · Invariance of the maximum likelihood estimator · Wishart distribution · Sampling law of the normal mean and scatter · Null distribution of Hotelling’s · Two-sample Hotelling test · Likelihood of a multivariate normal sample