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Let us consider a simple statistical model $\{f_{\theta}\}$ where $\theta\in U$, an open subset of $\mathbb{R}$. Let $X_1,\dots,X_n$ be sample drawn from $f_{\theta}$. I know, under some regularity assumptions, that $$ n \text{Var}_{\theta}\left(\hat{\theta}_{MLE}(X_1,\dots.X_n)\right)\to 1/I_{\theta},$$ where $I_{\theta}$ is the Fisher information. Can someone provide a proof (or a link to a reference) to the above result? In the books I have, I could find only proof of the following (relevant) result which is actually called asymptotic normality: $$\sqrt{n}\left(\hat{\theta}_{MLE}(X_1,\dots.X_n)-\theta\right)\stackrel{d}{\to} \mathcal{N}(0,1/I_{\theta}).$$ Thank you.

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By hypothesis, $\sqrt{n}\hat\theta_n=\sqrt{n}\theta+Z_n$ where $Z_n$ converges in distribution to a centered normal random variable $Z$ with variance $1/I_\theta$, and you are asking whether one can deduce that $E[Z_n^2]\to E[Z^2]$. Obviously, some supplementary hypothesis is needed, which, by uniform integrability, might be that $E[|Z_n|^3]$ is bounded uniformly in $n$.

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  • $\begingroup$ I was expecting a proof which is independent of the route of asymptotic normality proof. Anyway thank you for the answer. $\endgroup$
    – Ashok
    Jan 3, 2014 at 4:11

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