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 Le Cam (1960).

A sequence of parametric statistical models { P_n,θ: θ ∈ Θ } is said to be locally asymptotically normal (LAN) at θ if there exist matrices r_n and I_θ and a random vector Δ_n,θ ~ N(0, I_θ) such that, for every converging sequence h_n → h,

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 ${\displaystyle P_{\!n,\theta +r_{n}^{-1}h_{n}}}$ and ${\displaystyle P_{n,\theta }}$ are contiguous.

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