Я пытаюсь доказать, что наблюдаемая информационная матрица, оцененная по слабо непротиворечивой оценке максимального правдоподобия (MLE), является слабо непротиворечивой оценкой ожидаемой информационной матрицы. Это широко цитируемый результат, но никто не дает ссылку или доказательство (я исчерпал, я думаю, первые 20 страниц результатов Google и мои учебники статистики)!
Используя слабо согласованную последовательность MLE, я могу использовать слабый закон больших чисел (WLLN) и теорему о непрерывном отображении, чтобы получить желаемый результат. Однако я считаю, что теорема о непрерывном отображении не может быть использована. Вместо этого я думаю, что нужно использовать единый закон больших чисел (ULLN). Кто-нибудь знает ссылку, которая имеет доказательство этого? У меня есть попытка ULLN, но пока я ее опущу для краткости.
Я прошу прощения за длину этого вопроса, но обозначения должны быть введены. Обозначения как следующие (мое доказательство в конце).
Предположим, что у нас есть выборка случайных величин { Y 1 , … , Y N }
I ( θ ) = - E θ [ H θ ( log f ( ˜ Y | θ ) ]
где H θ
I N ( θ ) = N ∑ i = 1 I y i ( θ ) ,
где I y i = - E θ [ H θ ( log f ( Y i | θ ) ]
J ( θ ) = - H θ ( log f ( y | θ )
(некоторые люди требуют матрица оценивается в & thetas , но некоторые этого не делают). Выборочная наблюдаемая информационная матрица имеет вид;
J N ( θ ) = ∑ N i = 1 J y i ( θ )
где J y i ( θ ) = - H θ ( log f ( y i | θ )
I can prove convergence in probability of the estimator N−1JN(θ)
Now (JN(θ))rs=−∑Ni=1(Hθ(logf(Yi|θ))rs
Any help on this would be greatly appreciated.
Ответы:
I guess directly establishing some sort of uniform law of large numbers is one possible approach.
Here is another.
We want to show that JN(θMLE)NP⟶I(θ∗)JN(θMLE)N⟶PI(θ∗) .
(As you said, we have by the WLLN that JN(θ)NP⟶I(θ)JN(θ)N⟶PI(θ) . But this doesn't directly help us.)
One possible strategy is to show that |I(θ∗)−JN(θ∗)N|P⟶0.
and
|JN(θMLE)N−JN(θ∗)N|P⟶0
If both of the results are true, then we can combine them to get |I(θ∗)−JN(θMLE)N|P⟶0,
which is exactly what we want to show.
The first equation follows from the weak law of large numbers.
The second almost follows from the continuous mapping theorem, but unfortunately our function g()g() that we want to apply the CMT to changes with NN :
our gg is really gN(θ):=JN(θ)NgN(θ):=JN(θ)N . So we
cannot use the CMT.
(Comment: If you examine the proof of the CMT on Wikipedia, notice that the set BδBδ they define in their proof for us now
also depends on nn . We essentially need some sort of equicontinuity at θ∗θ∗
over our functions gN(θ)gN(θ) .)
Fortunately, if you assume that the family G={gN|N=1,2,…}G={gN|N=1,2,…}
is stochastically equicontinuous at θ∗θ∗ , then it immediately
follows that for θMLEP⟶θ∗θMLE⟶Pθ∗ ,
|gn(θMLE)−gn(θ∗)|P⟶0.
(See here: http://www.cs.berkeley.edu/~jordan/courses/210B-spring07/lectures/stat210b_lecture_12.pdf for a definition of stochastic equicontinuity at θ∗θ∗ , and a proof of the above fact.)
Therefore, assuming that GG is SE at θ∗θ∗ , your desired result holds
true and the empirical Fisher information converges to the population Fisher information.
Now, the key question of course is, what sort of conditions do you need to impose on GG to get SE?
It looks like one way to do this is to establish a Lipshitz condition
on the entire class of functions GG (see here: http://econ.duke.edu/uploads/media_items/uniform-convergence-and-stochastic-equicontinuity.original.pdf ).
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The answer above using stochastic equicontinuity works very well, but here I am answering my own question by using a uniform law of large numbers to show that the observed information matrix is a strongly consistent estimator of the information matrix , i.e. N−1JN(ˆθN(Y))a.s.⟶I(θ0)N−1JN(θ^N(Y))⟶a.s.I(θ0) if we plug-in a strongly consistent sequence of estimators. I hope it is correct in all details.
We will use IN={1,2,...,N}IN={1,2,...,N} to be an index set, and let us temporarily adopt the notation J(˜Y,θ):=J(θ)J(Y~,θ):=J(θ) in order to be explicit about the dependence of J(θ)J(θ) on the random vector ˜YY~ . We shall also work elementwise with (J(˜Y,θ))rs(J(Y~,θ))rs and (JN(θ))rs=∑Ni=1(J(Yi,θ))rs(JN(θ))rs=∑Ni=1(J(Yi,θ))rs , r,s=1,...,kr,s=1,...,k , for this discussion. The function (J(⋅,θ))rs(J(⋅,θ))rs is real-valued on the set Rn×Θ∘Rn×Θ∘ , and we will suppose that it is Lebesgue measurable for every θ∈Θ∘θ∈Θ∘ . A uniform (strong) law of large numbers defines a set of conditions under which
supθ∈Θ|N−1(JN(θ))rs−Eθ[(J(Y1,θ))rs]|=supθ∈Θ|N−1∑Ni=1(J(Yi,θ))rs−(I(θ))rs|a.s⟶0(1)supθ∈Θ∣∣N−1(JN(θ))rs−Eθ[(J(Y1,θ))rs]∣∣=supθ∈Θ∣∣N−1∑Ni=1(J(Yi,θ))rs−(I(θ))rs∣∣⟶a.s0(1)
The conditions that must be satisfied in order that (1) holds are (a) Θ∘Θ∘ is a compact set; (b) (J(˜Y,θ))rs(J(Y~,θ))rs is a continuous function on Θ∘Θ∘ with probability 1; (c) for each θ∈Θ∘θ∈Θ∘ (J(˜Y,θ))rs(J(Y~,θ))rs is dominated by a function h(˜Y)h(Y~) , i.e. |(J(˜Y,θ))rs|<h(˜Y)|(J(Y~,θ))rs|<h(Y~) ; and
(d) for each θ∈Θ∘θ∈Θ∘ Eθ[h(˜Y)]<∞Eθ[h(Y~)]<∞ ;. These conditions come from Jennrich (1969, Theorem 2).
Now for any yi∈Rnyi∈Rn , i∈INi∈IN and θ′∈S⊆Θ∘, the following inequality obviously holds
|N−1∑Ni=1(J(yi,θ′))rs−(I(θ′))rs|≤supθ∈S|N−1∑Ni=1(J(yi,θ))rs−(I(θ))rs|.(2)
Suppose that {ˆθN(Y)} is a strongly consistent sequence of estimators for θ0, and let ΘN1=BδN1(θ0)⊆K⊆Θ∘ be an open ball in Rk with radius δN1→0 as N1→∞, and suppose K is compact. Then since ˆθN(Y)∈ΘN1 for N sufficiently large enough we have P[limN{ˆθN(Y)∈ΘN1}]=1 for sufficiently large N. Together with (2) this implies
P[limN→∞{|N−1∑Ni=1(J(Yi,ˆθN(Y)))rs−(I(ˆθN(Y)))rs|≤supθ∈ΘN1|N−1∑Ni=1(J(Yi,θ))rs−(I(θ))rs|}]=1.(3)
Now ΘN1⊆Θ∘ implies conditions (a)-(d) of Jennrich (1969, Theorem 2) apply to ΘN1. Thus (1) and (3) imply
P[limN→∞{|N−1∑Ni=1(J(Yi,ˆθN(Y)))rs−(I(ˆθN(Y)))rs|=0}]=1.(4)
Since (I(ˆθN(Y)))rsa.s.⟶I(θ0) then (4) implies that N−1(JN(ˆθN(Y)))rsa.s.⟶(I(θ0))rs. Note that (3) holds however small ΘN1 is, and so the result in (4) is independent of the choice of N1 other than N1 must be chosen such that ΘN1⊆Θ∘. This result holds for all r,s=1,...,k, and so in terms of matrices we have N−1JN(ˆθN(Y))a.s.⟶I(θ0).
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