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Let $\theta$ be a parameter and $\hat{\theta}$ the plug-in estimate, I need a proof of the following lemma, as given in [1], p. 173, in the form of a reference or of a direct argument:

Percentile interval lemma. Suppose the transformation $\hat{\phi}=m(\phi)$ perfectly normalizes the distribution of $\hat{\theta}$: $\hat{\phi}\sim\mathcal{N}(\phi,c^2)$ for some standard deviation $c$. Then the percentile interval based on $\hat{\theta}$ equals: $(m^{-1}(\hat{\phi}-z^{(1-\alpha)}c),\, m^{-1}(\hat{\phi}-z^{(\alpha)}c))$.

Reference

[1] Efron, B. and Tibshirani, R.J. (1994) An Introduction to the Bootstrap (1st ed.) p. 173, Chapman and Hall/CRC

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