Past exam of the mathematics course of the University of Cambridge / 2026 / ib / Paper 1 / 6H (Statistics)

Factorisation of the likelihood $(1-\theta {)}^{-n}{1}_{\{\theta \le X_{(1)}\}}$ shows that $T=X_{(1)}$ is sufficient: conditional on it, the sample law is independent of $\theta$.

The Rao–Blackwell theorem says that for convex loss, conditioning any estimator on a sufficient statistic cannot increase risk; for squared error it preserves the mean and weakly reduces variance.

Since $E{X}_{(1)}=\theta +(1-\theta )/(n+1)$, take $\hat{\theta }=((n+1)X_{(1)}-1)/n$. It is unbiased, and the supplied variance formula gives $\mathrm{var}\hat{\theta }=(1-\theta {)}^2/[n(n+2)]$, which is its MSE.