“Local asymptotic normality”的意思、由来-开放百科全书

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Local asymptotic normality

释义

Definition
Example
See also
Notes
References

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 {{harvtxt|Le Cam|1960}}.

Definition

{{Technical|reason=notation should be explained - for example, what is θ here?|date=September 2010}}

A sequence of parametric statistical models {{nowrap|{ 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 {{nowrap|Δ_n,θ ~ N(0, I_θ)}} such that, for every converging sequence {{math|h_n → h}},^[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 {{nowrap|{ X₁, X₂, …, X_n }}} is an iid sample, where each X_i has density function {{nowrap|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 {{nowrap|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:

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

{{Cite book

{{Cite journal

{{Cite book

| last = van der Vaart | first = A.W.
| title = Asymptotic statistics
| year = 1998
| publisher = Cambridge University Press
| isbn = 978-0-521-78450-4
| ref = harv{{Refend}}{{DEFAULTSORT:Local Asymptotic Normality}}

1 : Asymptotic theory (statistics)

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Definition

Example

See also

Notes

References