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I understand that pivotal quantities are usually found for finding confidence intervals via inversion. However, I am wondering if there is an intuition behind a pivotal quantity having to depend on our data and parameters $(X, \theta)$ while having a distribution that is free of the parameters. Is it so that we can isolate the parameter $\theta$ by itself in the confidence interval? If so, what property of the pivotal quantity allows this?

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Yes. Essentially, the use of pivotality is that we obtain a statement of the sort $$1-\alpha=P\{\alpha/2\text{%-quantile}\leq\text{pivotal quantity}(X,\theta)\leq(1-\alpha/2)\text{%-quantile}\}$$ which holds for any $\theta$ so that we do not need to attach a $\theta$ to $P$ (as in $P_\theta$). We may then, as you say, rearrange the statements within the curly braces until we have isolated $\theta$. Since the equality still says $$1-\alpha=P\{...\}$$ we have derived a statement which holds for any $\theta$.

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    $\begingroup$ I just want to note that it is also useful to have one that removes dependence on nuisance parameters such as the variance in in the the case of the t test for the mean of a normal distribution. Also it is not always possible to construct a pivotal quantity. $\endgroup$ Dec 31, 2016 at 17:00

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