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Rao-Blackwell Theorem

The Rao-Blackwell theorem states that if T is an estimator of a parameter theta and S is a sufficient statistic for theta, then the conditional expectation

T^*=E[T|S]

is at least as good as T under every loss that is a convex function of the estimate and for which the expectation values exist. For the squared-error loss underlying mean square error, this gives

E[(T^*-theta)^2]<=E[(T-theta)^2].

If T is an unbiased estimator, then T^* is also an unbiased estimator and has no larger variance. The theorem therefore provides a systematic way to improve an estimator by conditioning it on a sufficient statistic.

See also

Conditional Expectation, Estimator, Mean Square Error, Sufficient Statistic, Variance

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References

Blackwell, D. "Conditional Expectation and Unbiased Sequential Estimation." Ann. Math. Stat. 18, 105-110, 1947. https://doi.org/10.1214/aoms/1177730497.Rao, C. R. "Information and Accuracy Attainable in the Estimation of Statistical Parameters." Bull. Calcutta Math. Soc. 37, 81-91, 1945.

Cite this as:

Weisstein, Eric W. "Rao-Blackwell Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Rao-BlackwellTheorem.html

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