Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

9.9 LEHMANN–SCHEFFÉ THEOREM

The transformation described in Section 9.8 does not necessarily generate the UMVU estimator. It guarantees only that the variance of the improved estimator will not exceed that of the original statistic T. The theorem presented in this section describes how to obtain the UMVU estimator. We first give a result concerning the uniqueness of estimators.

Theorem 9.6. Let T be a complete sufficient statistic for θ. If there exists an unbiased estimator g(T) for , then the estimator is unique.

Proof. The theorem is proved by contradiction. Consider another estimator h(T) that is unbiased and differs from g(T) in probability: . Since both estimators are unbiased:

(9.123)

which implies h(T)−g(T) = 0 because T is complete. This contradicts the assumption , so h(T) must be the same as g(T), which means that g(T) is unique.

Theorem 9.7 brings together the following properties of an estimator: sufficiency, completeness, unbiasedness, and minimum variance.

Theorem 9.7