F.6 CRAMÉR–RAO INEQUALITY
The Cramér–Rao inequality provides a lower bound on the variance of an estimator; it is also called the Cramér-Rao lower bound (CRLB). The CRLB is useful as a performance measure for biased and unbiased estimators of the parameters of a distribution, as described in Chapter 9. We provide a derivation for an unbiased estimator, and state the results for a biased estimator and then for a vector of parameters.
Theorem F.11 (Cramér–Rao inequality). Let T be an unbiased estimator for the function 
of scalar parameter 
where 
are iid samples with pdf 
. Let T = t be a specific value of the estimator for outcomes 
, and assume the following regularity conditions:
- Differentiation with respect to the parameter:
(F.46)
- Differentiation and integration exchange I:
(F.47)
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