📂Mathematical Statistics

Efficient Estimator

Definition ¹

Let’s say $Y$ is an unbiased estimator for the parameter $\theta$.

1. The efficiency of estimator $Y$ with respect to the Cramér-Rao lower bound $\text{RC}$ is defined as: $$ {{ \text{RC} } \over { \operatorname{Var} (Y) }} $$
2. An estimator with efficiency $1$ is called an Efficient Estimator.

Description

Cramér-Rao Inequality: $$ \operatorname{Var} (Y) \ge {{ \left[ k’(\theta) \right]^{2} } \over { n I (\theta) }} = \text{RC} $$

It is obvious that efficiency cannot exceed $1$ according to the inequality above.

Saying an estimator is efficient perfectly matches intuition; if the actual variance equals the Cramér-Rao lower bound, it means its theoretical variance is the smallest possible—it points to the parameter $\theta$ within the narrowest possible interval.

Difference from the Best Unbiased Estimator

At first glance, it seems similar to the Best Unbiased Estimator, but an efficient estimator has its variance exactly lowered to the Cramér-Rao bound, making it the theoretically optimal unbiased estimator. In contrast, the best unbiased estimator doesn’t need to theoretically beat every other unbiased estimator or lower the variance to the theoretical limit; it merely needs to surpass all other unbiased estimators. Being the best does not guarantee an efficiency of $1$, nor does failing to minimize variance to the theoretical lower limit preclude an estimator from being the best unbiased estimator.

While an efficient estimator is a best unbiased estimator, the converse is not necessarily true.