Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 2 / 29K / d / Solution

The risk of ${\overline{X}}_n$ is constantly $1/n$. Suppose an estimator $\delta$ dominates it, and put
$m(\theta )={\mathbb{E}}_{\theta }\delta$ and $b(\theta )=m(\theta )-\theta$. The biased Cramer-Rao bound, using total information $n$, givesDomination would implyfor every real $\theta$. The standard differential-inequality argument shows that the only globally defined differentiable solution is $b\equiv 0$: a nonzero value forces $b^{\prime }$ to retain a sign and magnitude that makes $b$ leave the permitted bounded interval in one time direction.

Thus $m(\theta )=\theta$. The Cramer--Rao inequality now gives
${\mathrm{Var}}_{\theta }\delta \ge 1/n$, so domination forces equality everywhere. Equality in Cramer--Rao makes the centred estimator proportional to the Gaussian score:and hence $\delta ={\overline{X}}_n$ almost surely. No strict improvement exists, so ${\overline{X}}_n$ is admissible.