Page 391 - Probability and Statistical Inference

368 7. Point Estimation

which can be rewritten as

by virtue of the Cauchy-Schwarz inequality or the covariance inequality (Theo-
rems 3.9.5-3.9.6). Recall that Y is the sum of n iid random variables and thus
in view of (7.5.7), we obtain

Now the inequality (7.5.1) follows by combining (7.5.8) and (7.5.9). "
Remark 7.5.1 It is easy to see that the CRLB given by the rhs of the
inequality in (7.5.1) would be attained by the variance of the estimator T for
all θ ∈ Θ if and only if we can conclude the strict equality in (7.5.8), that is if
and only if the statistic T and the random variable Y are linearly related w.p.1.
That is, the CRLB will be attained by the variance of T if and only if

with some fixed real valued functions a(.) and b(.).
Remark 7.5.2 By combining the CRLB from (7.5.1) and the expression
for the information I (θ), defined in (6.4.1), we can immediately restate the
X
Cramér-Rao inequality under the same standard assumptions as follows:

where I (θ) is the information about the unknown parameter θ in one single
X1
observation X .
1
We will interchangeably use the form of the Cramér-Rao inequality
given by (7.5.1) or (7.5.11). The CRLB would then correspond to
the expressions found on the rhs of either of these two equations.

Since the denominator in the CRLB involves the information, we would
rely heavily upon some of the worked out examples from Section 6.4.
Example 7.5.1 Let X , ..., X be iid Poisson(λ) where λ(> 0) is the
n
1
unknown parameter. Let us consider T(λ) = λ so that T′(λ) = 1. Now,
is an unbiased estimator of λ Vλ( ) = n λ and . Recall from the Ex-
-1
ample 7.4.4 that we could not claim that was the UMVUE of λ. Can
we now claim that is the UMVUE of λ? In (6.4.2) we find I (λ) = λ -1
X1
-1
and hence from the rhs of (7.5.11) we see that the CRLB = 1/(nλ ) = λ/n