Page 392 - Probability and Statistical Inference

7. Point Estimation 369

which coincides with the expression for That is, the estimator attains
the smallest possible variance among all unbiased estimators of ?. Then,
must be the UMVUE of λ.!
Example 7.5.2 Let X , ..., X be iid Bernoulli(p) where 0 < p < 1 is the
1
n
unknown parameter. Let us consider T(p) = p so that T′(p) = 1. Now, is an
unbiased estimator of p and Recall from the Example
7.4.1 that we could not claim that was the UMVUE of p. Can we now claim
that is the UMVUE of p? Let us first derive the expression for I (p).
X 1
Observe that

-1
so that log f(x; p) = xp - (1 - x)(1 - p) = (x - p){p(1 - p)} . Hence, one
-1
-1
evaluates I (p) as
X1

Thus from (7.5.11), we have the CRLB = 1/(n{p(1 - p)} ) = p(1 - p)/n which
-1
coincides with the expression for That is, attains the smallest pos-
sible variance among all unbiased estimators of p. Then, must be the UMVUE
of p. !

Example 7.5.3 Suppose that X , ..., X are iid N(µ, σ ) where µ is un-
2
n
1
known but σ is known. Here we have −∞ < µ < ∞, 0 < σ < ∞ and χ = ℜ. We
2
wish to estimate T(µ) = µ unbiasedly. Consider which is obviously an unbi-
ased estimator of µ. Is the UMVUE of µ? Example 7.4.6 was not decisive
in this regard. In (6.4.3) we find I (µ) = σ so that from (7.5.11) we have
-2
X1
--2
the CRLB = 1/(n σ ) = σ /n which coincides with the expression for
2
Then, must be the UMVUE of µ. !
In these examples, we thought of a “natural” unbiased estimator of the
parametric function of interest and this estimator’s variance happened to co-
incide with the CRLB. So, in the end we could claim that the estimator we
started with was in fact the UMVUE of T(µ). The reader has also surely noted
that these UMVUE’s agreed with the Rao-Blackwellized versions of some of
the naive initial unbiased estimators.
We are interested in unbiased estimators of T(µ). We found the
Rao-Blackwellized version W of an initial unbiased estimator T.
But, the variance of this improved estimator W may not attain
the CRLB. Look at the following example.
Example 7.5.4 (Example 7.4.5 Continued) Suppose that X , ..., X n
1
are iid Poisson(λ) where 0 < λ < ∞ is unknown with n ≥ 2. We wish to