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 Template:Harvtxt and is fundamental in the treatment of estimator and test efficiency.

Definition

Template:Technical A sequence of parametric statistical models Template:Nowrap} is said to be locally asymptotically normal (LAN) at θ if there exist matrices r_n and I_θ and a random vector Template:Nowrap such that, for every converging sequence Template:Math,

$\ln \frac{d{P}_{n,\theta +r_n^{-1}{h}_n}}{d{P}_{n,\theta }}=h^{\prime }{\mathrm{\Delta }}_{n,\theta }-\frac{1}{2}{h}^{\prime }{I}_{\theta }h+o_{P_{n,\theta }}(1),$

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:

$\ln \frac{d{P}_{n,\theta +r_n^{-1}{h}_n}}{d{P}_{n,\theta }}\stackrel{d}{\to }𝒩(-\frac{1}{2}{h}^{\prime }{I}_{\theta }h,h^{\prime }{I}_{\theta }h).$

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 Template:Nowrap} is an iid sample, where each X_i has density function Template:Nowrap. The likelihood function of the model is equal to

$p_{n,\theta }(x_1,\dots ,x_n;\theta )=\prod _{i=1}^{n}f(x_i,\theta ).$

If f is twice continuously differentiable in θ, then

$\begin{array}{cc}\ln p_{n,\theta +\delta \theta }& \approx \ln p_{n,\theta }+\delta {\theta }^{\prime }\frac{\partial \ln p_{n,\theta }}{\partial \theta }+\frac{1}{2}\delta {\theta }^{\prime }\frac{{\partial }^2\ln p_{n,\theta }}{\partial \theta \partial {\theta }^{\prime }}\delta \theta \\ & =\ln p_{n,\theta }+\delta {\theta }^{\prime }\sum _{i=1}^n\frac{\partial \ln f(x_i,\theta )}{\partial \theta }+\frac{1}{2}\delta {\theta }^{\prime }\left[\sum _{i=1}^n\frac{{\partial }^2\ln f(x_i,\theta )}{\partial \theta \partial {\theta }^{\prime }}\right]\delta \theta .\end{array}$

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

$\ln \frac{p_{n,\theta +h/\sqrt{n}}}{p_{n,\theta }}=h^{\prime }\left(\frac{1}{\sqrt{n}}\sum _{i=1}^n\frac{\partial \ln f(x_i,\theta )}{\partial \theta }\right)-\frac{1}{2}{h}^{\prime }(\frac{1}{n}\sum _{i=1}^n-\frac{{\partial }^2\ln f(x_i,\theta )}{\partial \theta \partial {\theta }^{\prime }})h+o_p(1).$

By the central limit theorem, the first term (in parentheses) converges in distribution to a normal random variable Template:Nowrap, whereas by the law of large numbers the expression in second parentheses converges in probability to I_θ, which is the Fisher information matrix:

$I_{\theta }=\mathrm{E}\left[-\frac{{\partial }^2\ln f(X_i,\theta )}{\partial \theta \partial {\theta }^{\prime }}\right]=\mathrm{E}\left[\right(\frac{\partial \ln f(X_i,\theta )}{\partial \theta }\left)\right(\frac{\partial \ln f(X_i,\theta )}{\partial \theta }{)}^{\prime }].$

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.

See also

• Asymptotic distribution

References

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