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 Le Cam (1960) 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,
ln d P n , θ + r n − 1 h n d P n , θ = h ′ Δ n , θ − 1 2 h ′ I θ h + o P n , θ ( 1 ) , {\displaystyle \ln {\frac {dP_{\!n,\theta +r_{n}^{-1}h_{n}}}{dP_{n,\theta }}}=h'\Delta _{n,\theta }-{\frac {1}{2}}h'I_{\theta }\,h+o_{P_{n,\theta }}(1),}
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 little 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:
ln d P n , θ + r n − 1 h n d P n , θ → d N ( − 1 2 h ′ I θ h , h ′ I θ h ) . {\displaystyle \ln {\frac {dP_{\!n,\theta +r_{n}^{-1}h_{n}}}{dP_{n,\theta }}}\ \ {\xrightarrow {d}}\ \ {\mathcal {N}}{\Big (}{-{\tfrac {1}{2}}}h'I_{\theta }\,h,\ h'I_{\theta }\,h{\Big )}.}
The sequences of distributions P n , θ + r n − 1 h n {\displaystyle P_{\!n,\theta +r_{n}^{-1}h_{n}}} and P n , θ {\displaystyle P_{n,\theta }} 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
p n , θ ( x 1 , … , x n ; θ ) = ∏ i = 1 n f ( x i , θ ) . {\displaystyle p_{n,\theta }(x_{1},\ldots ,x_{n};\,\theta )=\prod _{i=1}^{n}f(x_{i},\theta ).}
If f is twice continuously differentiable in θ, then
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