Rao–Blackwell theorem

In statistics, the Rao–Blackwell theorem, sometimes referred to as the Rao–Blackwell–Kolmogorov theorem, is a result which characterizes the transformation of an arbitrarily crude estimator into an estimator that is optimal by the mean-squared-error criterion or any of a variety of similar criteria.

The Rao–Blackwell theorem states that if g(X) is any kind of estimator of a parameter θ, then the conditional expectation of g(X) given T(X), where T is a sufficient statistic, is typically a better estimator of θ, and is never worse. Sometimes one can very easily construct a very crude estimator g(X), and then evaluate that conditional expected value to get an estimator that is in various senses optimal.

The theorem is named after Calyampudi Radhakrishna Rao and David Blackwell. The process of transforming an estimator using the Rao–Blackwell theorem is sometimes called Rao–Blackwellization. The transformed estimator is called the Rao–Blackwell estimator.

Definitions

In other words, a sufficient statistic T(X) for a parameter θ is a statistic such that the conditional distribution of the data X, given T(X), does not depend on the parameter θ.

The theorem

Mean-squared-error version

One case of Rao–Blackwell theorem states:

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

In other words

\operatorname{E}((\delta_1(X)-\theta)^2)\leq 
       \operatorname{E}((\delta(X)-\theta)^2).\,\!

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(Y2) cannot be less than [E(Y)]2. That inequality is a case of Jensen's inequality, although it may also be shown to follow instantly from the frequently mentioned fact that

 0 \leq \operatorname{Var}(Y) = \operatorname{E}((Y-\operatorname{E}(Y))^2) = 
               \operatorname{E}(Y^2)-(\operatorname{E}(Y))^2.\,\!

Convex loss generalization

The more general version of the Rao–Blackwell theorem speaks of the "expected loss" or risk function:

\operatorname{E}(L(\delta_1(X)))\leq \operatorname{E}(L(\delta(X)))\,\!

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

Properties

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 Rao–Blackwell estimator is no worse than the original estimator. In practice, however, the improvement is often enormous.

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 X1, ..., Xn of phone calls that arrived during n successive one-minute periods are observed. It is desired to estimate the probability eλ that the next one-minute period passes with no phone calls.

An extremely crude estimator of the desired probability is

\delta_0=\left\{\begin{matrix}1 & \text{if}\ X_1=0, \\ 0 & \text{otherwise,}\end{matrix}\right.

i.e., it estimates this probability to be 1 if no phone calls arrived in the first minute and zero otherwise. Despite the apparent limitations of this estimator, the result given by its Rao–Blackwellization is a very good estimator.

The sum

 S_n = \sum_{i=1}^n X_{i} = X_1+\cdots+X_n\,\!

can be readily shown to be a sufficient statistic for λ, i.e., the conditional distribution of the data X1, ..., Xn, depends on λ only through this sum. Therefore, we find the Rao–Blackwell estimator

\delta_1=\operatorname{E}(\delta_0\mid S_n=s_n).

After doing some algebra we have

\begin{align}
\delta_1 &= \operatorname{E} \left (\mathbf{1}_{\{X_1=0\}} \Bigg| \sum_{i=1}^n X_{i} = s_n \right ) \\
&= P \left (X_{1}=0 \Bigg| \sum_{i=1}^n X_{i} = s_n \right ) \\
&= P \left (X_{1}=0, \sum_{i=2}^n X_{i} = s_n \right ) \times P \left (\sum_{i=1}^n X_{i} = s_n \right )^{-1} \\
&= e^{-\lambda}\frac{\left((n-1)\lambda\right)^{s_n}e^{-(n-1)\lambda}}{s_n!} \times \left (\frac{(n\lambda)^{s_n}e^{-n\lambda}}{s_n!} \right )^{-1} \\
&= \frac{\left((n-1)\lambda\right)^{s_n}e^{-n\lambda}}{s_n!} \times \frac{s_n!}{(n\lambda)^{s_n}e^{-n\lambda}} \\
&= \left(1-\frac{1}{n}\right)^{s_n}
\end{align}

Since the average number of calls arriving during the first n minutes is nλ, one might not be surprised if this estimator has a fairly high probability (if n is big) of being close to

\left(1-{1 \over n}\right)^{n\lambda}\approx e^{-\lambda}.

So δ1 is clearly a very much improved estimator of that last quantity. In fact, since Sn is complete and δ0 is unbiased, δ1 is the unique minimum variance unbiased estimator by the Lehmann–Scheffé theorem.

Idempotence

Rao–Blackwellization is an idempotent operation. Using it to improve the already improved estimator does not obtain a further improvement, but merely returns as its output the same improved estimator.

Completeness and LehmannScheffé minimum variance

If the conditioning statistic is both complete and sufficient, and the starting estimator is unbiased, then the RaoBlackwell estimator is the unique "best unbiased estimator": see Lehmann–Scheffé theorem.

See also

References

External links

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