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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).

kjetil b halvorsen
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Tim
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  • What do you mean when you say "*What about the null has more than one distributions?*" - can you give an example? – Glen_b Apr 30 '13 at 01:15
  • @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 Apr 30 '13 at 01:24
  • Your explanation is not any clearer than before - indeed it appears to be a restatement in the same phrasing. Please give an explicit example. – Glen_b Apr 30 '13 at 01:27
  • @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 Apr 30 '13 at 01:29
  • @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 Apr 30 '13 at 01:45
  • Hang on, you're talking about a pair of point-nulls (or rather, I mean a single null consisting of two disjoint point-values), is that right? – Glen_b Apr 30 '13 at 03:24
  • @Glen_b: here is [a snapshot](http://i.stack.imgur.com/vhore.png) from the book – Tim Apr 30 '13 at 03:42
  • Thanks for the snapshot. What I read there makes perfect sense but doesn't seem correspond to what you seemed to me to be saying; that may be more a matter of the way you expressed it. Did you simply mean the part about composite nulls? – Glen_b Apr 30 '13 at 06:52
  • @Glen_b: Yes, By "the null has more than one distributions", I mean the null is composite. – Tim Apr 30 '13 at 12:02

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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

kjetil b halvorsen
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