La matrice di informazioni osservate è uno stimatore coerente della matrice di informazioni prevista?

Sto cercando di dimostrare che la matrice di informazioni osservate valutata allo stimatore della massima verosimiglianza debolmente coerente (MLE) è uno stimatore debolmente coerente della matrice di informazioni attesa. Questo è un risultato ampiamente citato ma nessuno fornisce un riferimento o una prova (ho esaurito penso che le prime 20 pagine dei risultati di Google e i miei libri di testo delle statistiche)!

Usando una sequenza debolmente coerente di MLE posso usare la legge debole di grandi numeri (WLLN) e il teorema di mappatura continua per ottenere il risultato che desidero. Tuttavia, credo che il teorema della mappatura continua non possa essere usato. Penso invece che debba essere utilizzata la legge uniforme dei grandi numeri (ULLN). Qualcuno sa di un riferimento che ne ha una prova? Ho un tentativo all'ULLN ma per ora lo ometto per brevità.

Mi scuso per la lunghezza di questa domanda, ma la notazione deve essere introdotta. La notazione è come gente (la mia prova è alla fine).

Supponiamo di avere un campione iid di variabili casuali $\{Y_1,\ldots,Y_N\}$ con densità $f(\tilde{Y}|\theta)$ , dove $\theta\in\Theta\subseteq\mathbb{R}^{k}$ (qui $\tilde{Y}$ è solo una variabile casuale generica con la stessa densità come uno qualsiasi dei membri del campione). Il vettore $Y=(Y_1,\ldots,Y_N)^{T}$ è il vettore di tutti i vettori campione in cui $Y_{i}\in\mathbb{R}^{n}$ per tutti$i=1,\ldots,N$ . Il vero valore del parametro di densità è$\theta_{0}$ e θ N ( Y ) è debolmente coerente stimatore di massima verosimiglianza (MLE) di θ 0 . Fatte salve le condizioni di regolarità, la matrice Informazioni Fisher può essere scritta come$\hat{\theta}_{N}(Y)$$\theta_{0}$

dove ${H}_{\theta}$ è la matrice hessiana. L'equivalente del campione è

dove $I_{y_i}=-E_\theta \left[H_{\theta}(\log f(Y_{i}|\theta)\right]$ . La matrice di informazioni osservate è;

$J(\theta) = -H_\theta(\log f(y|\theta)$ ,

(alcune persone chiedono la matrice è valutata a θ , ma alcuni non lo fanno). La matrice di informazioni osservate di esempio è;$\hat{\theta}$

$J_N(\theta)=\sum_{i=1}^N J_{y_i}(\theta)$

dove J y i ( θ ) = - H θ ( log f ( y i | θ ) .$J_{y_i}(\theta)=-H_\theta(\log f(y_{i}|\theta)$

I can prove convergence in probability of the estimator $N^{-1}J_N(\theta)$ to $I(\theta)$, but not of $N^{-1}J_{N}(\hat{\theta}_N(Y))$ to $I(\theta_{0})$. Here is my proof so far;

Ora ( J N ( θ ) ) r s = - ∑ N i = 1 ( H θ ( log f ( Y i | θ ) ) r s è l'elemento ( r , s ) di J N ( θ ) , per ogni r , s = 1 , … , $(J_{N}(\theta))_{rs}=-\sum_{i=1}^N (H_\theta(\log f(Y_i|\theta))_{rs}$$(r,s)$$J_N(\theta)$$r,s=1,\ldots,k$ . Se il campione è iid, allora dalla legge debole di grandi numeri (WLLN), la media di questi riepiloghi converge in probabilità a - E θ [ ( H θ ( log f ( Y 1 | θ ) ) r s ] = ( I Y 1 ( θ ) ) r s = ( I ( θ ) ) r s . Quindi N - 1 ( J N ( θ $-E_{\theta}[(H_\theta(\log f(Y_{1}|\theta))_{rs}]=(I_{Y_1}(\theta))_{rs}=(I(\theta))_{rs}$) r s P → ( I ( θ ) ) r s per tutto r , s = 1 , … , k , e quindi N - 1 J N ( θ ) P → I ( θ ) . Purtroppo non possiamo concludere semplicemente N - 1 J N ( θ N ( Y ) ) P → I ( $N^{-1}(J_N(\theta))_{rs}\overset{P}{\rightarrow}(I(\theta))_{rs}$$r,s=1,\ldots,k$$N^{-1}J_N(\theta)\overset{P}{\rightarrow}I(\theta)$$N^{-1}J_{N}(\hat{\theta}_N(Y))\overset{P}{\rightarrow}I(\theta_0)$ by using the continuous mapping theorem since $N^{-1}J_{N}(\cdot)$ is not the same function as $I(\cdot)$.

Any help on this would be greatly appreciated.

$\newcommand{\convp}{\stackrel{P}{\longrightarrow}}$

I guess directly establishing some sort of uniform law of large numbers is one possible approach.

Here is another.

We want to show that $\frac{J^N(\theta_{MLE})}{N} \convp I(\theta^*)$.

(As you said, we have by the WLLN that $\frac{J^N(\theta)}{N} \convp I(\theta)$. But this doesn't directly help us.)

One possible strategy is to show that

and

If both of the results are true, then we can combine them to get

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 $g()$ that we want to apply the CMT to changes with $N$: our $g$ is really $g_N(\theta) := \frac{J^N(\theta)}{N}$. So we cannot use the CMT.

(Comment: If you examine the proof of the CMT on Wikipedia, notice that the set $B_\delta$ they define in their proof for us now also depends on $n$. We essentially need some sort of equicontinuity at $\theta^*$ over our functions $g_N(\theta)$.)

Fortunately, if you assume that the family $\mathcal{G} = \{g_N | N=1,2,\ldots\}$ is stochastically equicontinuous at $\theta^*$, then it immediately follows that for $\theta_{MLE} \convp \theta^*$,

(See here: http://www.cs.berkeley.edu/~jordan/courses/210B-spring07/lectures/stat210b_lecture_12.pdf for a definition of stochastic equicontinuity at $\theta^*$, and a proof of the above fact.)

Therefore, assuming that $\mathcal{G}$ is SE at $\theta^*$, 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 $\mathcal{G}$ to get SE? It looks like one way to do this is to establish a Lipshitz condition on the entire class of functions $\mathcal{G}$ (see here: http://econ.duke.edu/uploads/media_items/uniform-convergence-and-stochastic-equicontinuity.original.pdf ).

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. $N^{-1}J_{N}(\hat{\theta}_{N}(Y))\overset{a.s.}{\longrightarrow}I(\theta_{0})$ if we plug-in a strongly consistent sequence of estimators. I hope it is correct in all details.

We will use $I_{N}=\{1,2,...,N\}$ to be an index set, and let us temporarily adopt the notation $J(\tilde{Y},\theta):=J(\theta)$ in order to be explicit about the dependence of $J(\theta)$ on the random vector $\tilde{Y}$. We shall also work elementwise with $(J(\tilde{Y},\theta))_{rs}$ and $(J_{N}(\theta))_{rs}=\sum\nolimits_{i=1}^{N}(J(Y_{i},\theta))_{rs}$, $r,s=1,...,k$, for this discussion. The function $(J(\cdot,\theta))_{rs}$ is real-valued on the set $\mathbb{R}^{n}\times\Theta^{\circ}$, and we will suppose that it is Lebesgue measurable for every $\theta\in\Theta^{\circ}$. A uniform (strong) law of large numbers defines a set of conditions under which

$\underset{\theta\in\Theta}{\text{sup}}\left|N^{-1}(J_{N}(\theta))_{rs}-E_{\theta}\left[(J(Y_{1},\theta))_{rs}\right]\right|=\nonumber\\ \hspace{60pt}\underset{\theta\in\Theta}{\text{sup}}\left|N^{-1}\sum\nolimits_{i=1}^{N}(J(Y_{i},\theta))_{rs}-(I(\theta))_{rs}\right|\overset{a.s}{\longrightarrow}0\hspace{100pt}(1)$

The conditions that must be satisfied in order that (1) holds are (a) $\Theta^{\circ}$ is a compact set; (b) $(J(\tilde{Y},\theta))_{rs}$ is a continuous function on $\Theta^{\circ}$ with probability 1; (c) for each $\theta\in \Theta^{\circ}$ $(J(\tilde{Y},\theta))_{rs}$ is dominated by a function $h(\tilde{Y})$, i.e. $|(J(\tilde{Y},\theta))_{rs}|<h(\tilde{Y})$; and (d) for each $\theta\in \Theta^{\circ}$ $E_{\theta}[h(\tilde{Y})]<\infty$;. These conditions come from Jennrich (1969, Theorem 2).

Now for any $y_{i}\in\mathbb{R}^{n}$, $i\in I_{N}$ and $\theta'\in S\subseteq\Theta^{\circ}$, the following inequality obviously holds

$\left|N^{-1}\sum\nolimits_{i=1}^{N}(J(y_{i},\theta'))_{rs}-(I(\theta'))_{rs}\right|\leq\underset{\theta\in S}{\text{sup}}\left|N^{-1}\sum\nolimits_{i=1}^{N}(J(y_{i},\theta))_{rs}-(I(\theta))_{rs}\right|.\hspace{50pt}(2)$

Suppose that $\{\hat{\theta}_{N}(Y)\}$ is a strongly consistent sequence of estimators for $\theta_{0}$, and let $\Theta_{N_{1}}=B_{\delta_{N_{1}}}(\theta_{0})\subseteq K\subseteq \Theta^{\circ}$ be an open ball in $\mathbb{R}^{k}$ with radius $\delta_{N_{1}}\rightarrow 0$ as $N_{1}\rightarrow\infty$, and suppose $K$ is compact. Then since $\hat{\theta}_{N}(Y)\in \Theta_{N_{1}}$ for $N$ sufficiently large enough we have $P[\underset{N}{\text{lim}}\{\hat{\theta}_{N}(Y)\in\Theta_{N_{1}}\}]=1$ for sufficiently large $N$. Together with (2) this implies

$P\left[\underset{N\rightarrow\infty}{\text{lim}}\left\{\left|N^{-1}\sum\nolimits_{i=1}^{N}(J(Y_{i},\hat{\theta}_{N}(Y)))_{rs}-(I(\hat{\theta}_{N}(Y)))_{rs}\right|\leq\right.\right.\nonumber\\ \hspace{40pt}\left.\left.\underset{\theta\in\Theta_{N_{1}}}{\text{sup}}\left|N^{-1}\sum\nolimits_{i=1}^{N}(J(Y_{i},\theta))_{rs}-(I(\theta))_{rs}\right|\right\}\right]=1.\hspace{100pt}(3)$

Now $\Theta_{N_{1}}\subseteq\Theta^{\circ}$ implies conditions (a)-(d) of Jennrich (1969, Theorem 2) apply to $\Theta_{N_{1}}$. Thus (1) and (3) imply

$P\left[\underset{N\rightarrow\infty}{\text{lim}}\left\{\left|N^{-1}\sum\nolimits_{i=1}^{N}(J(Y_{i},\hat{\theta}_{N}(Y)))_{rs}-(I(\hat{\theta}_{N}(Y)))_{rs}\right|=0\right\}\right]=1.\hspace{100pt}(4)$

Since $(I(\hat{\theta}_{N}(Y)))_{rs}\overset{a.s.}{\longrightarrow}I(\theta_{0})$ then (4) implies that $N^{-1}(J_{N}(\hat{\theta}_{N}(Y)))_{rs}\overset{a.s.}{\longrightarrow}(I(\theta_{0}))_{rs}$. Note that (3) holds however small $\Theta_{N_{1}}$ is, and so the result in (4) is independent of the choice of $N_{1}$ other than $N_{1}$ must be chosen such that $\Theta_{N_{1}}\subseteq \Theta^{\circ}$. This result holds for all $r,s=1,...,k$, and so in terms of matrices we have $N^{-1}J_{N}(\hat{\theta}_{N}(Y))\overset{a.s.}{\longrightarrow}I(\theta_{0})$.