9.8 RAO–BLACKWELL THEOREM
The Rao–Blackwell (RB) theorem describes a method for transforming a relatively poor estimator of θ, which is usually straightforward to find, into an improved estimator with lower variance. In order to prove the theorem, we need Jensen's inequality in Appendix F:
(9.111) 
where X is a random variable defined on an open interval with cdf fX(x), and g(x) is a convex function (see Appendix B). Although the Rao–Blackwell theorem applies to any convex function as used in Jensen's inequality, we present the theorem for the variance of an estimator.
Theorem 9.5 (Rao–Blackwell). Let 
be a set of statistics that are jointly sufficient for function 
of parameter θ. Define another statistic T that is an unbiased estimator of 
. Then statistic 
has the following properties.
is a function of and thus is sufficient.- so that TRB is an unbiased estimator of .
- for all θ.
- var[ ...
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