In statistics, the

**Rao-Blackwell theorem**describes a technique that can transform an absurdly crude estimator into an estimator that is optimal by the mean-squared-error criterion or any of a variety of similar criteria. (Pronunciation:

*Rao*rhymes with "cow".)

Table of contents |

2 The theorem 3 Example 4 Idempotence of the Rao-Blackwell process 5 When is the Rao-Blackwell estimator the best possible? |

## Some prerequisite definitions

- An estimator is an
*observable*random variable (i.e. a statistic) used for estimating some*unobservable*quantity. For example, one may be unable to observe the average height of*all*male students at the University of X, but one may observe the heights of a random sample of 40 of them. The average height of those 40--the "sample average"--may be used as an estimator of the unobservable "population average". - 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 depend on any of the*unobservable*quantities such as the mean or standard deviation of the whole population from which the data*X*was taken. In the most frequently cited examples, the "unobservable" quantities are parameters that parametrize a known family of probability distributions according to which the data are distributed. - A
**Rao-Blackwell estimator**δ_{1}(*X*) of an unobservable quantity θ is the conditional expected value E(δ(*X*) |*T*(*X*)) of some estimator δ(*X*) given a sufficient statistic*T*(*X*). Call δ(*X*) the**"original estimator"**and δ_{1}(*X*) the**"improved estimator"**. It is important that the improved estimator be*observable*, i.e., that it not depend on θ. Generally, the conditional expected value of one function of these data given another function of these data*does*depend on θ, but the very definition of sufficiency given above entails that this one does not. - The
*mean squared error*of an estimator is the expected value of the square of its deviation from the unobservable quantity being estimated.

## The theorem

One case of Rao-Blackwell theorem states:

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

The essential tools of the proof besides the definition above are the law of total expectation and the fact that for any random variable *Y*, E(*Y*^{2}) cannot be less than [E(*Y*)]^{2}. That inequality is a case of Jensen's inequality, although in a statistics course it may be shown to follow instantly from the frequently mentioned fact that

*L*may be any convex function. For the proof of the more general version, Jensen's inequality cannot be dispensed with.

The improved estimator is unbiased if and only if the original estimator is unbiased, as may be seen at once by using the law of total expectation. The theorem holds regardless of whether biased or unbiased estimators are used.

The theorem seems very weak: it says only that the allegedly improved estimator is no worse than the original estimator. In practice, however, the improvement is often enormous, as an example can show.

## Example

Phone calls arrive at a switchboard according to a Poisson process at an average rate of λ per minute. This rate is not observable, but the numbers of phone calls that arrived during *n* successive one-minute periods are observed. It is desired to estimate the probability *e*^{−λ} that the next one-minute period passes with no phone calls. The answer given by Rao-Blackwell may perhaps be unexpected.

A *extremely* crude estimator of the desired probability is

The sum

*conditional*distribution of the data

*X*

_{1}, ...,

*X*

_{n}, given this sum, does not depend on λ. Therefore, we find the Rao-Blackwell estimator

*X*

_{1}+ ... +

*X*

_{n}of calls arriving during the first

*n*minutes is

*n*λ, one might not be surprised if this estimator has a fairly high probability (if

*n*is big) of being close to

_{1}is clearly a very much improved estimator of that last quantity.

## Idempotence of the Rao-Blackwell process

In case the sufficient statistic is also a complete statistic, i.e., one which "admits no unbiased estimator of zero", the Rao-Blackwell process is idempotent, i.e., using it to improve the already improved estimator does not do so, but merely returns as its output the same improved estimator.

## When is the Rao-Blackwell estimator the best possible?

If the improved estimator is both unbiased and complete, then the Lehmann-Scheffé theorem implies that it is the unique "best unbiased estimator."