Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 29L (Principles of Statistics)

The maximum-likelihood estimator is any measurable maximizer
${\hat{\theta }}_n\in \arg {\max }_{\theta }{\prod }_{i=1}^{n}f(X_i;\theta )$.

Under regularity,

The likelihood isso ${\hat{\theta }}_n=X_{(1)}$. Regular asymptotics fail because the support $[\theta ,\infty )$ depends on $\theta$ and the likelihood maximum lies on its boundary.

Write $X_i=\theta +E_i$ with independent rate-one exponentials. Their minimum is exponential of rate $n$, sofor every $n$. Thus $\alpha =1$ and the nondegenerate limit is rate-one exponential.

The MLE error is rate-$n$ exponential, henceSince ${\tilde{\theta }}_n={\bar{X}}_n-1$ is unbiased with variance $1/n$,The boundary-based MLE is asymptotically much more efficient.