Source: Wikipedia, the free encyclopedia.
In
regular parametric model
.
The notion of local asymptotic normality was introduced by Le Cam (1960) and is fundamental in the treatment of estimator and test efficiency .[1]
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
hn → h ,
[2]
ln
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{\displaystyle \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
converge in distribution
to a normal random variable whose mean is equal to minus one half the variance:
ln
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{\displaystyle \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
,
θ
+
r
n
−
1
h
n
{\displaystyle P_{\!n,\theta +r_{n}^{-1}h_{n}}}
P
n
,
θ
{\displaystyle P_{n,\theta }}
contiguous .[2]
Example The most straightforward example of a LAN model is an iid model whose likelihood is twice continuously differentiable. Suppose { X 1 , X 2 , …, Xn } is an iid sample, where each Xi has density function f (x , θ )
p
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n
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θ
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∏
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f
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{\displaystyle 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
ln
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≈
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[
∑
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{\displaystyle {\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
δ
θ
=
h
/
n
{\displaystyle \delta \theta =h/{\sqrt {n}}}
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{\displaystyle \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
Fisher information matrix
:
I
θ
=
E
[
−
∂
2
ln
f
(
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i
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∂
θ
∂
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=
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[
(
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∂
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{\displaystyle 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.
See also Notes References
Ibragimov, I.A.; Has’minskiĭ, R.Z. (1981). Statistical estimation: asymptotic theory . Springer-Verlag. . 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. .
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da20e3a17ff279c0f91a2c2e89448b7654bd0e3f)
![{\displaystyle 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 ]}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/64ff1762959d7543772d8ce17c04e91a80c2f878)