# Show that there is no efficient estimator for the variance of a normal distribution using properties of the exponential family

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:

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.