# 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 a rescaling of the parameter. An important example when the local asymptotic normality holds is in the case of iid sampling from a regular parametric model.

The notion of local asymptotic normality was introduced by Le Cam (1960).

## 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 hnh,[1]

$\ln \frac{dP_{\!n,\theta+r_n^{-1}h_n}}{dP_{n,\theta}} = h'\Delta_{n,\theta} - \frac12 h'I_\theta\,h + o_{P_{n,\theta}}(1),$

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:

$\ln \frac{dP_{\!n,\theta+r_n^{-1}h_n}}{dP_{n,\theta}}\ \ \xrightarrow{d}\ \ \mathcal{N}\Big( {-\tfrac12} h'I_\theta\,h,\ h'I_\theta\,h\Big).$

The sequences of distributions $P_{\!n,\theta+r_n^{-1}h_n}$ and $P_{n,\theta}$ are contiguous.[1]

### 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

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

If f is twice continuously differentiable in θ, then

\begin{align} \ln p_{n,\theta+\delta\theta} &\approx \ln p_{n,\theta} + \delta\theta'\frac{\partial \ln p_{n,\theta}}{\partial\theta} + \frac12 \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} + \frac12 \delta\theta' \bigg[\sum_{i=1}^n\frac{\partial^2 \ln f(x_i,\theta)}{\partial\theta\,\partial\theta'} \bigg]\delta\theta . \end{align}

Plugging in δθ = h / √n, gives

$\ln \frac{p_{n,\theta+h/\sqrt{n}}}{p_{n,\theta}} = h' \Bigg(\frac{1}{\sqrt{n}} \sum_{i=1}^n\frac{\partial \ln f(x_i,\theta)}{\partial\theta}\Bigg) \;-\; \frac12 h' \Bigg( \frac1n \sum_{i=1}^n - \frac{\partial^2 \ln f(x_i,\theta)}{\partial\theta\,\partial\theta'} \Bigg) h \;+\; o_p(1).$

By the central limit theorem, the first term (in parentheses) converges in distribution to a normal random variable 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:

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

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.

### Notes

1. ^ a b van der Vaart (1998, pp. 103–104)

## References

• Ibragimov, I.A.; Has’minskiĭ, R.Z. (1981). Statistical estimation: asymptotic theory. Springer-Verlag. ISBN 0-387-90523-5.
• Le Cam, L. (1960). "Locally asymptotically normal families of distributions". University of California Publications in Statistics 3: 37–98.
• van der Vaart, A.W. (1998). Asymptotic statistics. Cambridge University Press. ISBN 978-0-521-78450-4.