I tried to find answers the questions below, but I could not get clear answers for them.
For a random sample of size n, $x_1, x_2, ..., x_n$ from a Normal distribution where $\sigma^2$ is unknown.
It is quite easy to derive answers with the MLE of $\sigma^2$
However, how to show the biasedness and consistency for maximum likelihood estimator of $\sigma$? I mean, not $\sigma^2$
Moreover, how to show the asymptotic distribution of the MLE $\sigma$?