Unbiased estimators and UMVU estimators

Let $T$ be an estimator of a real estimand $\psi$ in a statistical experiment. It is unbiased if it is integrable under every $P_{\theta }$ and ${\mathbb{E}}_{\theta }[T]=\psi (\theta )\text{for every }\theta \in \mathrm{\Theta }\mathrm{.}$ Equivalently, its bias from Bias and mean squared error is zero at every parameter. It is a finite-variance unbiased estimator if it is unbiased and square-integrable under every $P_{\theta }$.

A finite-variance unbiased estimator $T$ is uniformly minimum-variance unbiased (UMVU) for $\psi$ if, for every $\theta \in \mathrm{\Theta }$ and every other finite-variance unbiased estimator $U$ of the same estimand, ${\mathrm{Var}}_{\theta }(T)\le {\mathrm{Var}}_{\theta }(U)\mathrm{.}$ The same estimator must satisfy these inequalities throughout the parameter space; the definition asserts no existence of a UMVU estimator. Variance has the meaning in Moments, variance, and covariance on a probability space.