Question

It is well known that for iid random variables $X_1, \ldots, X_n$ with variance $\sigma^2$ that

$$\frac{1}{n-1} \sum_{i=1}^n (X_i - \overline X)^2$$

gives an unbiased estimator for $\sigma^2$, but

$$\sqrt{ \frac{1}{n-1} \sum_{i=1}^n (X_i - \overline X)^2 }$$

is not an unbiased estimator for $\sigma$.

Is there a way to formulate precisely and prove that there does not exist an unbiased estimator $f(X_1,\ldots,X_n)$ for $\sigma$ that works for all probability distributions?

Answer 1

Consider the case $n=2$. By scaling and translation we must have $f(x,y) = c |x-y|$ for some constant $c$. But for e.g. a Bernoulli distribution $E[c|X_1 - X_2|] = c p(1-p)$, and this can't be $\sigma = \sqrt{p(1-p)}$ for all $p$.