Given a sample $X_1,...,X_n$ from a pdf of the form $\frac{1}{\sigma} f((x-\theta) / \sigma)$, list at least five different pivotal quantities.
A pivotal quantity is a function of the sample that does not depend on unknown parameters. Here are five different pivotal quantities for a sample $X_1,...,X_n$ from a pdf of the form $\frac{1}{\sigma} f((x-\theta) / \sigma)$:
$T = \frac{\sqrt{n}(\bar{X} - \theta)}{S}$ - the Student's t-statistic with $n-1$ degrees of freedom
$Z = \frac{\sqrt{n}(\bar{X} - \theta)}{\sigma}$ - the standardized normal deviate
$U = \frac{\max{X_1,...,X_n} - \theta}{\sigma}$ - the standardized maximum.
I only think of these 3 cases. Any other cases? I don't think sample mean and sample variance are the correct answers.