Local asymptotic normality

In statistics, local asymptotic normality is a property of a sequence of statistical models, which allows this sequence to be asymptotically approximated by a normal location model, after a rescaling of the parameter. An important example when the local asymptotic normality holds is in the case of iid sampling from a regular parametric model.
The notion of local asymptotic normality was introduced by .
Definition.
A sequence of parametric statistical models } is said to be locally asymptotically normal (LAN) at "θ" if there exist matrices "rn" and "Iθ" and a random vector such that, for every converging sequence ,
where the derivative here is a Radon–Nikodym derivative, which is a formalised version of the likelihood ratio, and where "o" is a type of big O in probability notation. In other words, the local likelihood ratio must converge in distribution to a normal random variable whose mean is equal to minus one half the variance:
The sequences of distributions formula_3 and formula_4 are contiguous.
Example.
The most straightforward example of a LAN model is an iid model whose likelihood is twice continuously differentiable. Suppose } is an iid sample, where each "Xi" has density function . The likelihood function of the model is equal to
If "f" is twice continuously differentiable in "θ", then
Plugging in formula_7, gives
By the central limit theorem, the first term (in parentheses) converges in distribution to a normal random variable , whereas by the law of large numbers the expression in second parentheses converges in probability to "Iθ", which is the Fisher information matrix:
Thus, the definition of the local asymptotic normality is satisfied, and we have confirmed that the parametric model with iid observations and twice continuously differentiable likelihood has the LAN property.