Univariate t test as a template for T²
Section of Estimation and inference for the multivariate normal, part of Statistical learning
Univariate t test as a template for T²
| Type | Section |
| Taught in | Estimation and inference for the multivariate normal |
| Part | Statistical learning |
The shape of the result is already visible when p=1. For independent \operatorname{N}(\mu,\sigma^2) data, the test of H_0:\mu=\mu_0 against H_a:\mu\neq\mu_0 uses
t=\frac{\bar x-\mu_0}{s/\sqrt{n}}\sim t_{n-1}
under H_0, where t_{n-1} is the Student t distribution. Under H_0 the distribution of t does not depend on the unknown (\mu,\sigma^2). A quantity with that property is a pivot, so the critical value can be taken from the t_{n-1} table alone. At level \alpha the two-sided rule rejects H_0 when |t|>t_{\alpha/2;\,n-1}. Throughout this page the first subscript of a critical value is the upper-tail probability, and the remaining subscripts are degrees of freedom. The same point is the quantile of order 1-\alpha in the notation of Statistical inference. The algebra below is arranged so that the multivariate statistic reduces to the square of this t when p=1.
Used in
See also
One-sample t test · Hotelling’s T² test · 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