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 an appropriate rescaling of the parameter. An important example when the local asymptotic normality holds is in the case of i.i.d sampling from a regular parametric model. The notion of local asymptotic normality was introduced by and is fundamental in the treatment of estimator and test efficiency.

Definition

A sequence of parametric statistical models { Pn,θ: θ ∈ Θ } is said to be locally asymptotically normal (LAN) at θ if there exist matrices rn and Iθ and a random vector Δn,θ ~ N(0, Iθ) such that, for every converging sequence hn → 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 and are contiguous.

Example

The most straightforward example of a LAN model is an iid model whose likelihood is twice continuously differentiable. Suppose { X1, X2, …, Xn } is an iid sample, where each Xi has density function f(x, θ). The likelihood function of the model is equal to If f is twice continuously differentiable in θ, then Plugging in, gives By the central limit theorem, the first term (in parentheses) converges in distribution to a normal random variable Δθ ~ N(0, Iθ), 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.