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Let us consider the Gaussian model $\mathcal{N}(\mu,\sigma^2)$, where both $\mu$ and $\sigma$ are unknown. I have learnt that (for example, from Amari's information geometry book) the exponential families always have efficient estimators (the ones which achieve Cramer-Rao lower bound) for its parameters.

We know that $\bar{X_n}=\frac{1}{n}\sum_{i=1}^n X_i$ is an unbiased and efficient estimator of $\mu$.

Also $S_{n-1}^2=\frac{1}{n − 1}\sum_{i=1}^n(X_i − \bar{X_n})^2$ is an unbiased estimator for $\sigma^2$; however this is not efficient.

Question:
Would the limit as $n$ tends to $\infty$ of $S_{n-1}^2$ be an efficient estimator? If so, what is the limit? Does $\sigma^2$ have an efficient estimator at all? If not, doesn't this contradict the statement in the first paragraph?

More generally, when we say an exponential family has efficient estimators, is it meant that the efficient estimator need not exist for a finite sample but always exists in the asymptotic sense?

Can someone clarify this?

Note: If this or a related question has been discussed already, I request that someone please direct me to that question.

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    $\begingroup$ If $\mu$ is known, you should use it. See here $\endgroup$ – Glen_b Feb 19 '14 at 3:05
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    $\begingroup$ That would be a new question (your question specified known $\mu$). Note that there may not always be an unbiased estimator that achieves the CRLB. $\endgroup$ – Glen_b Feb 19 '14 at 4:16
  • $\begingroup$ @Glen_b: I have edited the question accordingly. $\endgroup$ – Kumara Feb 19 '14 at 5:44

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