We know that an unbiased estimator of the variance is: $$ \hat{\sigma}^2_{unbiased} = \frac{1}{n-1}\sum_{i=1}^n (x_i - \bar{x})^2$$
I was wondering, does it have the smallest possible standard error?
Does the biased (but consistent) estimator: $$ \hat{\sigma}^2_{biased} = \frac{1}{n}\sum_{i=1}^n (x_i - \bar{x})^2$$ have a lower standard error?
I know that the biased estimator of the variance is the maximum likelihood estimator (MLE) and that MLE estimators have the smallest possible variance in the set of well-behaved estimators. According to this result, what I would conclude here is that $\sigma^2_{biased}$ even though it is biased, has a lower standard error than the unbiased estimator.