Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 3 / 28J (Principles of Statistics)

The log-likelihood is, up to constants, $n\alpha \log \beta -\beta {\sum }_i{X}_i$, so the maximum-likelihood estimator is $\hat{\beta }=\alpha /\overline{X}$. By the central limit theorem,Applying the delta method to $g(x)=\alpha /x$ gives

Let $z_{1-\gamma /2}$ be the corresponding standard normal distribution quantile. Slutsky theorem permits replacing $\beta$ by the consistent estimator $\hat{\beta }$, giving the confidence intervalAfter standardization, the preceding convergence in distribution and continuity of the normal distribution show that its coverage tends to $1-\gamma$.

For bootstrapping, repeatedly sample $X_1^{\ast },\dots ,X_n^{\ast }$ with replacement from the empirical distribution of the observations and compute ${\hat{\beta }}^{\ast }=\alpha /{\overline{X}}^{\ast }$. Conditional on the data, estimate the $\gamma /2$ and $1-\gamma /2$ quantiles $q_l,q_u$ of $\sqrt{n}({\hat{\beta }}^{\ast }-\hat{\beta })$. The basic bootstrap interval isAs the number of resamples tends to infinity it estimates the conditional quantiles; bootstrap consistency and the continuous mapping theorem then give asymptotic coverage $1-\gamma$.