Page 383 - Probability and Statistical Inference

360 7. Point Estimation

just before we introduced the Rao-Blackwell Theorem. The possible values of
T are 0 or 1. Of course is sufficient for p. The domain space for
U is u = {0, 1, 2, ..., n}. Let us write for u ∈ u,

Next, observe that is Binomial(n,p) and is Binomial(n - 1, p).
Also X and are independently distributed. Thus, we can immediately
1
rewrite (7.4.3) as

That is, the Rao-Blackwellized version of the initial unbiased estimator X
1
turns out to be , the sample mean, even though T was indeed a very naive
and practically useless initial estimator of p. Now note that V [T] = p(1-p) and
p
V [W] = p(1 - p)/n so that V [W] < V [T] if n ≥ 2. When n = 1, the sufficient
p
p
p
statistic is X and so if one starts with T = X , then the final estimator obtained
1
1
through Rao-Blackwellization would remain X . That is, when n = 1,
1
we will not see any improvement over T through the Rao-Blackwellization
technique.!
Start with an unbiased estimator T of a parametric function T(θθ θθ θ).
The process of conditioning T given a sufficient (for θθ θθ θ) statistic U
is referred to as Rao-Blackwellization. The refined estimator W
is often called the Rao-Blackwellized version of T. This technique
is remarkable because one always comes up with an improved
unbiased estimator W for T(θθ θθ θ) except in situations where the initial
estimator T itself is already a function of the sufficient statistic U.
Example 7.4.2 (Example 7.4.1 Continued) Suppose that X , ..., X are
1
n
iid Bernoulli(p) where 0 < p < 1 is unknown, with n ≥ 2. Again, we wish
to estimate T(p) = p unbiasedly. Now consider a different initial estima-
tor T = ½(X + X ) and obviously T is an unbiased estimator of p. The
2
1
possible values of T are 0, ½ or 1 and again is a sufficient