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:

enter image description here

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.