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Let us say we have a random sample $X_1,...,X_n\sim F(\theta)$ and we do inference over $\theta$ and give a maximum likelihood estimate $\hat{\theta}$ and a confidence interval at the $\alpha$% given by $(\hat{l}_\theta,\hat{u}_\theta)$. Does invariance apply for the quantiles of the distribution? This is, is $\hat{F}^{-1}(\beta;\theta)=F^{-1}(\beta;\hat{\theta})$? Moreover, can you make a confidence interval for the quantile with the confidence interval of the estimate? Such as $(F^{-1}(\beta;\hat{l}_\theta),F^{-1}(\beta;\hat{u}_\theta))$. Assume that $F$ might not have a closed form inverse and that $\theta$ is one dimensional.

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