# 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 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 { 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,
    $\ln {\frac {dP_{\!n,\theta +r_{n}^{-1}h_{n}}}{dP_{n,\theta }}}=h'\Delta _{n,\theta }-{\frac {1}{2}}h'I_{\theta }\,h+o_{P_{n,\theta }}(1),$
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 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:
    $\ln {\frac {dP_{\!n,\theta +r_{n}^{-1}h_{n}}}{dP_{n,\theta }}}\ \ {\xrightarrow {d}}\ \ {\mathcal {N}}{\Big{-{\tfrac {1}{2}}}h'I_{\theta }\,h,\ h'I_{\theta }\,h{\Big )}.$
The sequences of distributions $P_{\!n,\theta +r_{n}^{-1}h_{n}}$ and $P_{n,\theta }$ are contiguous.

### Example

The most straightforward example of a LAN model is an iid model whose likelihood is twice continuously differentiable. Suppose { X$_1$, X$_2$, …, X$_n$ } is an iid sample, where each X$_i$ has density function f(x, θ). The likelihood function of the model is equal to
    $p_{n,\theta }(x_{1},\ldots ,x_{n};\,\theta )=\prod _{i=1}^{n}f(x_{i},\theta ).$
If f is twice continuously differentiable in θ, then
    ${\begin{aligned}\ln p_{n,\theta +\delta \theta }&\approx \ln p_{n,\theta }+\delta \theta '{\frac {\partial \ln p_{n,\theta }}{\partial \theta }}+{\frac {1}{2}}\delta \theta '{\frac {\partial ^{2}\ln p_{n,\theta }}{\partial \theta \,\partial \theta '}}\delta \theta \\&=\ln p_{n,\theta }+\delta \theta '\sum _{i=1}^{n}{\frac {\partial \ln f(x_{i},\theta )}{\partial \theta }}+{\frac {1}{2}}\delta \theta '{\bigg\sum _{i=1}^{n}{\frac {\partial ^{2}\ln f(x_{i},\theta )}{\partial \theta \,\partial \theta '}}{\bigg ]}\delta \theta .\end{aligned}}$
Plugging in $\delta \theta =h/{\sqrt {n}}$, gives
    $\ln {\frac {p_{n,\theta +h/{\sqrt {n}}}}{p_{n,\theta }}}=h'{\Bigg{\frac {1}{\sqrt {n}}}\sum _{i=1}^{n}{\frac {\partial \ln f(x_{i},\theta )}{\partial \theta }}{\Bigg )}\;-\;{\frac {1}{2}}h'{\Bigg{\frac {1}{n}}\sum _{i=1}^{n}-{\frac {\partial ^{2}\ln f(x_{i},\theta )}{\partial \theta \,\partial \theta '}}{\Bigg )}h\;+\;o_{p}(1).$
By the central limit theorem, the first term (in parentheses) converges in distribution to a normal random variable Δ$_θ$ ~ N(0, I$_θ$), whereas by the law of large numbers the expression in second parentheses converges in probability to I$_θ$, which is the Fisher information matrix:
    $I_{\theta }=\mathrm {E} {\bigg{-{\frac {\partial ^{2}\ln f(X_{i},\theta )}{\partial \theta \,\partial \theta '}}}{\bigg ]}=\mathrm {E} {\bigg{\bigg{\frac {\partial \ln f(X_{i},\theta )}{\partial \theta }}{\bigg )}{\bigg{\frac {\partial \ln f(X_{i},\theta )}{\partial \theta }}{\bigg )}'\,{\bigg ]}.$
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.