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Purpose: Construct a 95% confidence interval for $\theta$.

A common strategy is to construct a confidence interval for $\log(\theta)$ and then exponentiate.

Why is it valid?

My concern is that $E(\log(\theta)) \neq \log(E(\theta))$ and $Var(\log(\theta)) \neq \log(Var(\theta))$...

Do people use this back transformation because it has proven to work fine? What about other transformations (e.g. logit)?

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It's valid because of simple probability arithmetic:

Consider a $1-\alpha$ CI for $\theta$, $(l,u)$. Consider a monotonic increasing transformation $t$.

In all the cases where
$\theta<l$, $t(\theta)<t(l)$;
$\theta=l$, $t(\theta)=t(l)$;
$l<\theta<u$, $t(l)<t(\theta)<t(u)$;
$\theta=u$, $t(\theta)=t(u)$; and
$\theta>u$, $t(\theta)>t(u)$.

Thus any probability properties associated with the interval for $\theta$ will also hold for $t(\theta)$, when the ends of the interval are similarly transformed.

The point about expectation and variance is true, but irrelevant to the behaviour.

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