n this section we obtain a low bound for variance of an estimator under some
reasonable regularity assumptions.
We use the Cauchy
inequality
where the
is a joint distribution of the sample
We wish to apply the above relationship with
and
hence, we
set
and
We
obtain
Therefore,
The last result is called the Cramer-Rao low bound.
If the variables
are iid then
and the denominator may be
simplified
Hence,
Therefore, according to the last section the MLE asymptotically achieves the
low bound.
|
