Page 384 - Probability and Statistical Inference

7. Point Estimation 361

statistic for p. Then, E [T | U = u] = ½{E [X | U = u] + E [X |
p p 1 p 2
In other words, if one starts with ½(X + X )
1 2
as the initial unbiased estimator of p, then after going through the process of
Rao-Blackwellization, one again ends up with as the refined unbiased esti-
mator of p. Observe that V [T] = p(1 - p)/2 and so that
p
if n = 3. When n = 2, the sufficient statistic is T, and so if one
happens to start with T, then the final estimator obtained through the process
of Rao-Blackwellization would remain T. In other words, when n = 2, we will
not see any improvement over T through the Rao-Blackwellization technique.!
Example 7.4.3 (Example 7.4.1 Continued) Suppose that X , ..., X are iid
1
n
Bernoulli(p) where 0 < p < 1 is unknown, with n ≥ 2. We wish to estimate T(p)
= p(1 - p) unbiasedly. Consider T = X (1 - X ) which is an unbiased estimator
1
2
of T(p). The possible values of T are 0 or 1. Again, is the suffi-
cient statistic for p with the domain space u = {0, 1, 2, ..., n}. Let us denote
and then write for u ∈ u:

since X (1 - X ) takes the value 0 or 1 only. Thus, we express E [X (1 - X ) |
p
1
2
1
2
U = u] as
Next observe that is Binomial(n, p), is Binomial(n - 2, p), and
also that X , are independently distributed. Thus, we can re-
1
write (7.4.4) as

which is the same as That is, the Rao-Blackwellized ver-
sion of the initial unbiased estimator X (1 - X ) turns out to be n(n-
1
2
For the Bernoulli random samples, since the X’s are either
0 or 1, observe that the sample variance in this situation turns out to be

which coincides with the Rao-
Blackwellized version. !