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]

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:

The sequences of distributions and 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

If f is twice continuously differentiable in θ, then

Plugging in , gives

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:

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.

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See also

Notes

  1. 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.CS1 maint: ref=harv (link)
  • Le Cam, L. (1960). "Locally asymptotically normal families of distributions". University of California Publications in Statistics. 3: 37–98.CS1 maint: ref=harv (link)
  • van der Vaart, A.W. (1998). Asymptotic statistics. Cambridge University Press. ISBN 978-0-521-78450-4.CS1 maint: ref=harv (link)
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