In statistics, the Rao-Blackwell theorem is a way to improve an estimator.

• An estimator is an observable random variable, i.e. a statistic, used for estimating some unobservable quantity.

• A sufficient statistic T(X) is an observable random variable such that the conditional probability distribution of all observable data X given T(X) does not depent on any of the unobservable quantities such as the mean or standard deviation of the whole population from which the data X was taken.

• A Rao-Blackwell estimator of &delta₁(X) an unobservable quantity θ is the conditional expected value E(δ(X) | T(X)) of an estimator δ(X) given a sufficient statistic T(X). Call δ(X) the "original estimator" and δ₁(X) the "improved estimator".

The Rao-Blackwell theorem states:

The mean squared error of the Rao-Blackwell estimator does not exceed that of the original estimator.

In other words