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,^[2]

${\displaystyle \ln {\frac {dP_{\!n,\theta +r_{n}^{-1}h_{n}}}{dP_{n,\theta }}}=h'\Delta _{n,\theta }-{\frac {1}{2}}h'I_{\theta }\,h+o_{P_{n,\theta }}(1),}$ 

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 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:

${\displaystyle \ln {\frac {dP_{\!n,\theta +r_{n}^{-1}h_{n}}}{dP_{n,\theta }}}\ \ {\xrightarrow {d}}\ \ {\mathcal {N}}{\Big (}{-{\tfrac {1}{2}}}h'I_{\theta }\,h,\ h'I_{\theta }\,h{\Big )}.}$ 

The sequences of distributions ${\displaystyle P_{\!n,\theta +r_{n}^{-1}h_{n}}}$  and ${\displaystyle P_{n,\theta }}$  are contiguous.^[2]

Example edit

The most straightforward example of a LAN model is an iid model whose likelihood is twice continuously differentiable. Suppose { X₁, X₂, …, 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

${\displaystyle p_{n,\theta }(x_{1},\ldots ,x_{n};\,\theta )=\prod _{i=1}^{n}f(x_{i},\theta ).}$ 

If f is twice continuously differentiable in θ, then

${\displaystyle {\begin{aligned}\ln p_{n,\theta +\delta \theta }&\approx \ln p_{n,\theta }+\delta \theta '{\frac {\partial \ln p_{n,\theta }}{\partial \theta }}+{\frac {1}{2}}\delta \theta '{\frac {\partial ^{2}\ln p_{n,\theta }}{\partial \theta \,\partial \theta '}}\delta \theta \\&=\ln p_{n,\theta }+\delta \theta '\sum _{i=1}^{n}{\frac {\partial \ln f(x_{i},\theta )}{\partial \theta }}+{\frac {1}{2}}\delta \theta '{\bigg [}\sum _{i=1}^{n}{\frac {\partial ^{2}\ln f(x_{i},\theta )}{\partial \theta \,\partial \theta '}}{\bigg ]}\delta \theta .\end{aligned}}}$ 

Plugging in ${\displaystyle \delta \theta =h/{\sqrt {n}}}$ , gives

${\displaystyle \ln {\frac {p_{n,\theta +h/{\sqrt {n}}}}{p_{n,\theta }}}=h'{\Bigg (}{\frac {1}{\sqrt {n}}}\sum _{i=1}^{n}{\frac {\partial \ln f(x_{i},\theta )}{\partial \theta }}{\Bigg )}\;-\;{\frac {1}{2}}h'{\Bigg (}{\frac {1}{n}}\sum _{i=1}^{n}-{\frac {\partial ^{2}\ln f(x_{i},\theta )}{\partial \theta \,\partial \theta '}}{\Bigg )}h\;+\;o_{p}(1).}$ 

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:

${\displaystyle I_{\theta }=\mathrm {E} {\bigg [}{-{\frac {\partial ^{2}\ln f(X_{i},\theta )}{\partial \theta \,\partial \theta '}}}{\bigg ]}=\mathrm {E} {\bigg [}{\bigg (}{\frac {\partial \ln f(X_{i},\theta )}{\partial \theta }}{\bigg )}{\bigg (}{\frac {\partial \ln f(X_{i},\theta )}{\partial \theta }}{\bigg )}'\,{\bigg ]}.}$ 

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.

• Asymptotic distribution

Notes edit