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I want to prove the statement in the title using the following statement from Wikipedia:

it was proved that efficient estimation is possible only in an exponential family, and only for the natural parameters of that family.

We know that Normal distribution $\mathcal{N}(\mu,\sigma^2)$ belongs to the exponential family. Also from Wikipedia table we have:

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So natural parameters of the Normal family $\mathcal{N}(\mu,\sigma^2)$ are $\eta_1 = \frac{\mu}{\sigma^2}, \, \eta_2 = -\frac{1}{2 \sigma^2}$ and there must exist efficient estimators for them.

But I see that $\sigma^2$ doesn't coincide with $\eta_1$ or $\eta_2$. Is this the end of the proof? If no, what should I do next?

P.S. I know that there exist other ways to prove the statement but I want to prove it using the abovementioned general propertiy of the exponential families.

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