Theory of Statistics - George Mason University

Then we have
E(g(θ)T(X)) = E(T(X)E(g(θ)|X)) = E((T(X)) 2 ).
rT(PΘ) = E((T(X) − g(θ)|X)) 2 )
= E((T(X)) 2 ) + E((g(θ)) 2 ) − 2E(g(θ)T(X))
= 0.
4.3 Bayes Rules 351
Hence, by the condition of equation (3.84) we have the following theorem.
Theorem 4.8
Suppose T is an unbiased estimator. Then T is not a Bayes estimator under
squared-error loss.
Examples
There are two standard examples of Bayesian analyses that serve as models
for Bayes estimation under squared-error loss. These examples, Example 4.6
and 4.8, should be in the student’s bag of easy pieces. In both of these examples,
the prior is in a parametric conjugate family.
In this section, we also consider estimation using an improper prior (Example
4.9).
Example 4.6 (Continuation of Example 4.2) Estimation of the Binomial
Parameter with Beta Prior and a Squared-Error Loss
We return to the problem in which we model the conditional distribution of an
observable random variable X as a binomial(n, π) distribution, conditional on
π, of course. Suppose we assume π comes from a beta(α, β) prior distribution;
that is, we consider a random variable Π that has beta distribution. We wish
to estimate Π.
Let us choose the loss to be squared-error. In this case we know the risk is
minimized by choosing the estimate as δ(x) = E(Π|x), where the expectation
is taken wrt the distribution with density fΠ|x.
We recognize the posterior conditional distribution as a beta(x + α, n −
x + β), so we have the Bayes estimator for squared-error loss and beta prior
α + X
. (4.45)
α + β + n
We should study this estimator from various perspectives.
linear combination of expectations
First, we note that it is a weighted average of the mean of the prior
and the standard UMVUE. (We discuss the UMVUE for this problem in
Examples 5.1 and 5.5.)
Theory of Statistics c○2000–2013 James E. Gentle