Page 385 - Probability and Statistical Inference

362 7. Point Estimation

Example 7.4.4 Suppose that X , ..., X are iid Poisson(λ) where 0 < l < ∞
n
1
is unknown. We wish to estimate T(λ) = λ unbiasedly. Consider T = X which
1
is an unbiased estimator of λ and is a sufficient statistic for ?.
The domain space for U is u = {0, 1, 2, ..}. Now, for u ∈ U, conditionally
given U = u, the statistic T can take one of the possible values from the set T
= {0, 1, 2, ..., u}. Thus, for u ∈ u, we can write

Now, we have to find the expression for P {T = t | U = u} for all fixed u ∈ u
and t ∈ . Let us write λ

But observe that is Poisson(nλ) is Poisson((n - 1)λ), and also
that X , are independently distributed. Thus, from (7.4.6) we can
1
express P {T = t | U = u} as
λ

That is, the conditional distribution of T given U = u is Binomial(u, 1/n).
Hence, combining (7.4.5) and (7.4.7), we note that
In other words, the Rao-Blackwellized version of T is , the sample mean.!

We started with trivial unbiased estimators in the Examples 7.4.1
-7.4.4. In these examples, perhaps one could intuitively guess the
the improved unbiased estimator. The Rao-Blackwell Theorem did
not lead to any surprises here. The Examples 7.4.5 and 7.4.7 are
however, different because we are forced to begin with naive
unbiased estimators. Look at the Exercises 7.4.6 and
7.4.8 for more of the same.

Example 7.4.5 (Example 7.4.4 Continued) Suppose that X , ..., X n
1
are iid Poisson(λ) where 0 < λ < ∞ is unknown and n ≥ 2. We wish to
estimate T(λ) = e unbiasedly. Consider T = I(X = 0) which is an
-λ
1
unbiased estimator of T(λ) since E [T] = P {X = 0} = e . Consider
-λ
λ λ 1