Minimum-variance unbiased estimator

Remarks

Here $\operatorname{Var}_\theta$ is the variance when $X \sim P_\theta$.

Example: let $X_1, \dots, X_n$ be i.i.d. Bernoulli($p$) with $\Theta = (0, 1)$. The proportion $\bar{X}$ is unbiased with variance $p(1-p)/n$. The Fisher information of the sample is $n/(p(1-p))$, so the Cramér–Rao bound for unbiased estimators of $p$ is $p(1-p)/n$. The estimator $\bar{X}$ attains it at every $p$, so $\bar{X}$ is the UMVUE of $p$.

Non-example: for $n \ge 3$, $T' = (X_1 + X_2)/2$ is unbiased, but its variance $p(1-p)/2$ exceeds $p(1-p)/n$ for every $p$, so it fails the variance inequality.

The UMVUE need not exist, and an estimator that attains the Cramér–Rao bound need not exist even when the UMVUE does. If $S$ is a complete sufficient statistic and $T$ is unbiased, then $E(T \mid S)$ is the UMVUE; this is the Lehmann–Scheffé theorem.