Page 393 - Probability and Statistical Inference

370 7. Point Estimation

-λ
estimate T(λ) = e unbiasedly. In the Example 7.4.5, we had started with T =
I(X = 0) but its Rao-Blackwellized version was in fact W =
1
Does V (W) attain the CRLB? Recall that is Poisson(nλ) so that its
λ
mgf is given by

Let us use (7.5.13) with s = 2log(1 - n ) to claim that E [W ] = exp{nλ(e -
s
-1
2
λ
2
1)} = exp{nλ[(n-1/n) − 1]} = exp{−2λ + n λ}. Hence, we obtain
-1
-λ
Now, we have T’(λ) = -e and, from (6.4.2), I (λ) = λ . Utilizing (7.5.11) we
-1
X1
-2λ
-1
2
obtain the CRLB = { T’(λ)} /(nλ ) = n λe . Now, for x > 0, observe that e >
-1
x
1 + x. Hence, from (7.5.14) we obtain
In other words, V [W] does not attain the CRLB. !
λ
Question remains whether the estimator W in the Example 7.5.4
-λ
is the UMVUE of e . It is clear that the CRLB alone may not
point toward the UMVUE. Example 7.5.5 is also similar.
Example 7.5.5 (Example 7.4.8 Continued) Suppose that X , ..., X are iid
1
n
N(µ, σ ) where µ is unknown but σ is known with −∞ < µ < ∞, 0 < σ < ∞
2
2
χ
and = ℜ. We wish to estimate T(µ) = µ unbiasedly. In the Example 7.4.8,
2
we found the Rao-Blackwellized unbiased estimator
Let us first obtain the expression of the variance of the estimator W as fol-
lows:

The first term in (7.5.15) is evaluated next. Recall that = 0
and since
has N(µ, n σ ) distribution. Thus, we have
2
-1