Efficiency of an Estimator

What does it mean for an estimator to be efficient?

Answer

An estimator is said to be efficient if it has the smallest possible variance among all unbiased estimators of the same parameter.

In other words, its variance achieves the Cramér–Rao lower bound:

\[ \mathrm{Var}(\hat{\theta}) = \frac{1}{I(\theta)}, \]

where $I(\theta)$ is the Fisher information.

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