# Can pivot be used for testing

A pivotal quantity $Q(X, \theta)$ can be used to construct a confidence interval. I was wondering if it can be used to construct a test statistic and rejection region? In simpler cases involving a simple null hypothesis it is obvious how to use a pivot to construct a test statistic and rejection region. What about composite null hypotheses?

Pivotal quantities are fundamental to the construction of test statistics, as they allow the statistic to not depend on parameters – for example, Student's t-statistic is for a normal distribution with unknown variance (and mean).

• What do you mean when you say "What about the null has more than one distributions?" - can you give an example? Commented Apr 30, 2013 at 1:15
• @Glen_b: By "the null has more than one distributions", I mean the null hypothesis $H$ consists of more than one distributions of $X$.
– Tim
Commented Apr 30, 2013 at 1:24
• Your explanation is not any clearer than before - indeed it appears to be a restatement in the same phrasing. Please give an explicit example. Commented Apr 30, 2013 at 1:27
• @Glen_b: For example, the null $H$ is $\{ P_{\theta_1}, P_{\theta_2}\}$, where there are two possible distributions for $X$. If the null $H$ is $\{ P_{\theta_3}\}$, then there is only one possible distribution for $X$.
– Tim
Commented Apr 30, 2013 at 1:29
• @Glen_b: When $H$ consists only one distribution of $X$, $H$ is called simple. When $H$ consists more than one distributions of $X$, $H$ is called composite. These are terminologies from Bickel and Doksum's Mathematical Statistics.
– Tim
Commented Apr 30, 2013 at 1:45