Conditioning a biased estimator on a sufficient statistic

Question

I'm afraid my awareness of the Rao–Blackwell theorem has been limited to textbook accounts and exercises, and those deal only with its application to unbiased estimators. Maybe it's properly called the Rao–Blackwell theorem only when it's a statement about unbiased estimators. (?)

Is there anything of interest to be said about the use as an estimator, of the conditional expected value of a biased estimator given a sufficient statistic? Published results? Examples of interest?

Answer 1

If you Rao-Blackwellize a biased estimator, the bias does not change, and the variance potentially shrinks. The result doesn't change much.

Let $U$ be the estimator for $h(\theta)$. Let $b = E[U]-h(\theta)$ be the bias. Let $T$ be the sufficient statistic. Let $U^* = E[U\mid T]$ be the Rao-Blackwellized estimator.

The bias stays the same by the law of total expectations: \begin{align*} E[U^*] &= E[E[U\mid T]] \\ &= E[U] \\ &= b+h(\theta). \end{align*}

The variance potentially shrinks by the law of total variance \begin{align*} V[U] &= V[E(U\mid T)] + E[V(U\mid T)] \\ &= V[U^*] + E[V(U\mid T)] \\ &\ge V[U^*]. \end{align*}