Asymptotic consistency of MLE. Fisher's information number.

roposition

Let
be an iid sample from the population with the distribution
.
Assume that the distribution
is a smooth function of
for all possible
,
and let
be the MLE of
then
in distribution as the sample size
approaches infinity. Here
is a standard normal variable and the
is the Fisher's information
number

Proof

We introduce the log-likelihood
function
and consider the Taylor
expansion
Since
is the MLE we have
,
hence
The
is a sum of iid random variables. The mean of each
is zero. Indeed,
Hence, according to the CLT (
Central Limit
theorem
)
in distribution.
Similarly,
with some number
.
Since the expression
is not zero, we
obtain
Observe
that
Indeed,
Hence,