We will now prove Theorem 6 and Theorem 7. It is important to follow the steps involved to help your understanding of likelihood concepts.
Lemma 8: Multivariate Central Limit Theorem.
Suppose that is a -dimensional vector variable
with mean vector and variance-covariance matrix
with finite values on the diagonal. If is an IID sequence of vector random
variables having the same distribution as , and if
(meaning a vector componentwise sum) then
as
Lemma 9: If then .
First recall that if RVs are iid , then by definition.
Letting , we have
(vector linear combination of zero mean r.v. – using result at start of chapter)
(using variance result at start of chapter)
(vector) linear combination of normal r.v. is also normal
So .
Hence . ∎
Lemma 10: Asymptotic distribution of the true score.
Under the regularity conditions,
.
Asymptotically as , .
Example 4.1: Normal Data, ctd.
, so .
In this case,
and
so
Since
and
it follows that
Example 4.2: Gamma Distribution, ctd.
.
In this case
where as before and
It follows that
where the determinant is
Notice that in Example 4.4 , and in Example 4.4 , but this is the exception rather than the rule.