Rao–Blackwell, polishing a crude estimator

Question

You have a crude unbiased estimator of some parameter and also a sufficient statistic $S$ for that parameter. A colleague claims you can always improve the crude estimator, for free, using $S$ .

State the Rao–Blackwell theorem, prove the variance never increases, and give an example.

Accepted Answer

The construction. Let T be any unbiased estimator of 	heta and S a sufficient statistic. Define the Rao–Blackwellized estimator T^\star = \mathbb{E}[T \mid S]. Because S is sufficient, this conditional expectation does not depend on the unknown 	heta, so T^\star is a legitimate statistic you can actually compute. It stays unbiased. By the tower property, \mathbb{E}[T^\star] = \mathbb{E}\big[\mathbb{E}[T\mid S]\big] = \mathbb{E}[T] = 	heta. It never has larger variance. The law of total variance says \operatorname{Var}(T) = \underbrace{\operatorname{Var}\big(\mathbb{E}[T\mid S]\big)}{\opera…

Rao–Blackwell, polishing a crude estimator

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