Jensen's inequality
Rao–Blackwell theorem

If L is a convex function and G {\displaystyle {\mathfrak {G}}} a sub-sigma-algebra, then, from the conditional version of Jensen's inequality, we get

So if δ(X) is some estimator of an unobserved parameter θ given a vector of observables X; and if T (X) is a sufficient statistic for θ; then an improved estimator, in the sense of having a smaller expected loss L, can be obtained by calculating

the expected value of δ with respect to θ, taken over all possible vectors of observations X compatible with the same value of T (X) as that observed. Further, because T is a sufficient statistic, δ 1 (X) {\displaystyle \delta _{1}(X)} does not depend on θ, hence, becomes a statistic.

This result is known as the Rao–Blackwell theorem.
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