Pivotal quantity

Remarks

Example: let $X_1, \dots, X_n$ be i.i.d. $\mathcal{N}(\mu, \sigma^2)$ with $\theta = (\mu, \sigma^2)$ unknown, let $\bar X$ be the sample mean and $S$ the sample standard deviation. Then $T = \sqrt{n}(\bar X - \mu)/S$ has Student's t-distribution with $n - 1$ degrees of freedom for every $\theta$, so $T$ is a pivotal quantity. Non-example: in the same model, $\sqrt{n}(\bar X - \mu)$ is not pivotal, since its law $\mathcal{N}(0, \sigma^2)$ depends on $\sigma^2$. A pivotal quantity need not be a statistic, since it may involve $\theta$. If $P_\theta(a \le g(X, \theta) \le b) = 1 - \alpha$ for every $\theta$, then the set $\set{\theta : a \le g(X, \theta) \le b}$ is a confidence region of level $1 - \alpha$.