Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 3 / 28L (Principles of Statistics)

Let $q_u=G^{-1}(u)$. Since $\sqrt{n}(\hat{\theta }-\theta )\sim G$, an exact interval is $[\hat{\theta }-q_{1-\alpha /2}/\sqrt{n},\hat{\theta }-q_{\alpha /2}/\sqrt{n}]$.

A bootstrap sample consists of $n$ independent draws with replacement from the empirical distribution assigning mass $1/n$ to each observed $X_i$.

Replace the two $G$ quantiles by conditional bootstrap quantiles ${\hat{q}}_u$ in the interval from (a). Uniform convergence implies ${\hat{q}}_u\to q_u$ in probability at continuity points, so Slutsky and continuity of $G$ give coverage tending to $1-\alpha$.

Generate $B$ resamples, compute $\sqrt{n}({\hat{\theta }}^{\ast j}-\hat{\theta })$, and use their empirical $\alpha /2$ and $1-\alpha /2$ quantiles. Large $B$ approximates the otherwise unavailable conditional bootstrap distribution.