Minimum-variance unbiased estimator

Minimum variance unbiased estimator (MVUE or MVU estimator) is an estimator of parameters derived from estimation theory. The MVUE is only valid for unbiased estimators and results when the variance of the estimator is minimized for all values of the parameters. If the variance isn't minimized for all values of the parameters, then it's not the MVUE.

$\mathrm {mse} \left({\hat {theta}}\right)=\mathrm {E} \left[\left({\hat {\theta }}-\theta \right)^{2}\right]=\mathrm {var} \left({\hat {\theta }}\right)+\mathrm {bias} \left(\theta \right)^{2}$

now constricting the bias to zero

$\mathrm {mse} \left({\hat {\theta }}\right)=\mathrm {var} \left({\hat {\theta }}\right)$

and minimizing the MSE is equivalent to the MVUE since it is minimizing the variance of an unbiased estimator.