Past exam of the statistics course of the University of Oxford / 2021 / 1 / a / iv / Solution

The specified asymptotic relative efficiency isFor every fixed $\theta >0$ this lies strictly between zero and one. Prefer $T_2$ on grounds of asymptotic precision: it has the smaller leading asymptotic variance and attains the inverse Fisher information. Both are consistent estimators, by the law of large numbers and the continuous mapping theorem applied respectively to $\bar{X}$ and $A/n$. The efficiency approaches zero as $\theta \downarrow 0$ and one as $\theta \to \infty$.

This preference is an asymptotic comparison. In particular, it does not assert that $T_2$ is an unbiased estimator: since $A$ has a gamma distribution of shape $n$ and rate $\theta$, $\mathbb{E}{T}_2=n\theta /(n-1)$ for $n>1$.