Simultaneous confidence statements
Section of Estimation and inference for the multivariate normal, part of Statistical learning
Simultaneous confidence statements
| Type | Section |
| Taught in | Estimation and inference for the multivariate normal |
| Part | Statistical learning |
Invert the test, as with the one-sample region. The values of \boldsymbol{\delta}=\boldsymbol{\mu}_1-\boldsymbol{\mu}_2 that the level-\alpha test would not reject form the confidence set
\bigl(\bar{\mathbf{x}}^{(1)}-\bar{\mathbf{x}}^{(2)}-\boldsymbol{\delta}\bigr)^\top S^{-1} \bigl(\bar{\mathbf{x}}^{(1)}-\bar{\mathbf{x}}^{(2)}-\boldsymbol{\delta}\bigr) \le\frac{n_1+n_2}{n_1 n_2}\,T^2_{\alpha;\,p,\,n_1+n_2-p-1},
where
T^2_{\alpha;\,p,\,n_1+n_2-p-1} =\frac{(n_1+n_2-2)\,p}{n_1+n_2-p-1}\,F_{\alpha;\,p,\,n_1+n_2-p-1}.
The same critical value yields simultaneous intervals for every linear functional \mathbf{a}^\top\boldsymbol{\delta}, \mathbf{a}\in\mathbb{R}^p, and not only for the p coordinates:
\mathbf{a}^\top\bigl(\bar{\mathbf{x}}^{(1)}-\bar{\mathbf{x}}^{(2)}\bigr) \pm \sqrt{\mathbf{a}^\top S\mathbf{a}\cdot\frac{n_1+n_2}{n_1 n_2}\,T^2_{\alpha;\,p,\,n_1+n_2-p-1}}.
A coordinate interval is the case in which \mathbf{a} is a unit vector along one axis. The half-width uses the T^2 critical value rather than a single two-sample t critical value, because the intervals are required to cover \mathbf{a}^\top\boldsymbol{\delta} simultaneously for every \mathbf{a}, including the direction in which the two sample means differ most. Separate t intervals each control one coordinate and do not control a search over directions, so the simultaneous intervals are wider.
Used in
See also
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 · Null distribution of Hotelling’s · Two-sample Hotelling test · Likelihood of a multivariate normal sample