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 i.i.d 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 { 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 $h n \to h$ ,[1]

\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.[1]

Example

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

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.

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.

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[V-1] van der Vaart (1998, pp. 103–104)

Local asymptotic normality

Definition

Example

See also

Notes

References