Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics III / 7 / b

Use the gamma density and the normalization integral for shape $n-1$:The same argument, valid for $n\ge 3$, gives $\mathbb{E}[Y^{-2}]={\theta }^2/[(n-1)(n-2)]$.

The inverse moments in part i giveThus the bias is positive, and direct expansion of the mean squared error yieldsFor comparison, the variance is $n^2{\theta }^2/[(n-1{)}^2(n-2)]$; adding the squared bias gives the same answer by the bias-variance decomposition.

The central limit theorem givesWrite $z=z_{1-\alpha /2}$ for the indicated standard normal quantile. Inverting the approximate pivot inequality and substituting $Y/n=1/\hat{\theta }$ gives the approximate confidence intervalIts coverage is approximately $1-\alpha$ for large $n$. The intersection respects the positive rate parameter; asymptotically the lower endpoint is positive. This is an asymptotic interval, not an exact finite-sample confidence interval.