📂Mathematical Statistics

Best Unbiased Estimator, Minimum Variance Unbiased Estimator UMVUE

Definition ¹

Let us assume that parameter $\theta$ is given. If an unbiased estimator $W^{\ast}$ satisfies the following condition over all other unbiased estimators $W$, it is called the Best Unbiased Estimator or the Uniform Minimum Variance Unbiased Estimator (UMVUE). $$ \operatorname{Var}_{\theta} W^{\ast} \le \operatorname{Var}_{\theta} W \qquad , \forall \theta $$

Explanation

UMVUE is sometimes simply referred to as MVUE, dropping the initial Uniform part. The term UMVUE might be too lengthy, and while the expression “Best” fits quite well, “Minimum Variance” is also very intuitive and since “Best” is not exactly an academic term, the phrase Best Unbiased Estimator is rarely used across both Korean and English.

Difference from an Efficient Estimator

At first glance, it may seem similar to an efficient estimator, but an efficient estimator lowers its variance exactly to the Cramér-Rao bound, making it an unbiased estimator that theoretically cannot be improved upon, while the Best Unbiased Estimator doesn’t have to reach this theoretical limit but merely needs to surpass all other unbiased estimators. Striving for the best does not guarantee efficiency, defined as $1$, and not minimizing the variance to its theoretical lower limit does not preclude being the Best Unbiased Estimator.

Being an efficient estimator makes it the Best Unbiased Estimator, but the converse does not hold.