The concept of Local Asymptotic Normality (LAN) – introduced by Lucien LeCam – is one of the most important and fundamental ideas of the general asymptotic statistical theory. The LAN property is of particular importance in the asymptotic theory of testing, estimation and discriminant analysis. Many statistical models have got likelihood ratios which are locally asymptotic normal – that is the likelihood ratio processes of those models are asymptotically similar to those for the normal location parameter.
Let and be two sequences of probability measures on . Suppose there is a sequence , of sub -algebras of s.th. and . Let be the restriction of to and let be the Radon-Nikodym density taken on of the part of that is dominated by . Put
where and .
The logarithm of likelihood ratio
taken on is then
(LeCam 1986). Suppose that under the following conditions are satisfied
- L2: ,
- L3: , and
- L4: for some . then
Proof: Note that
where if . Note that
For some define
hence using L1
Note also that
Hence using L3, L4
Thus can be expressed as
Since we can use the dominated convergence theorem to have
which implies that
Note that is a martingale difference array and that
Note that is i) uniformly bounded in norm, ii) and iii) .
Hence by applying McLeish’s central limit theorem we finally prove that