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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).

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  • $\begingroup$ What do you mean when you say "What about the null has more than one distributions?" - can you give an example? $\endgroup$
    – Glen_b
    Commented Apr 30, 2013 at 1:15
  • $\begingroup$ @Glen_b: By "the null has more than one distributions", I mean the null hypothesis $H$ consists of more than one distributions of $X$. $\endgroup$
    – Tim
    Commented Apr 30, 2013 at 1:24
  • $\begingroup$ Your explanation is not any clearer than before - indeed it appears to be a restatement in the same phrasing. Please give an explicit example. $\endgroup$
    – Glen_b
    Commented Apr 30, 2013 at 1:27
  • $\begingroup$ @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 $X$. $\endgroup$
    – Tim
    Commented Apr 30, 2013 at 1:29
  • $\begingroup$ @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. $\endgroup$
    – Tim
    Commented Apr 30, 2013 at 1:45

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Confidence intervals can often be constructed from hypothesis tests, yes. (Some discussion and diverging opinions on this topic can be found here: https://andrewgelman.com/2013/06/24/why-it-doesnt-make-sense-in-general-to-form-confidence-intervals-by-inverting-hypothesis-tests/). That process is generally called inverting a confidence interval. Whether the confidence interval is constructed from a pivot or otherwise is immaterial.

Some posts about inversion: Intuition for why confidence intervals can be constructed by inverting tests, Confidence intervals derived from 'inverted hypothesis test', Inverting a hypothesis test: nitpicky detail

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