0
$\begingroup$

Given a random i.i.d. sample from a population with a finite variance $\sigma^2<\infty$, what estimator of $\sigma^2$ is optimal under absolute loss? $$ \arg\min_{\hat\sigma^{2}\in F}\mathbb{E}(|\hat\sigma^2-\sigma^2|) $$ where $F$ (for lack of a better symbol) is some sensible, broad class of variance estimators and the expectation is taken over the sampling distribution.

In case there is no optimal estimator, is there one that is somehow "good" or "natural"?

Improve this question
$\endgroup$
3
  • 1
    $\begingroup$ My first thought is, Median squared difference from sample mean adjusted for the difference between sample mean and population mean, though I do not have an idea on how to do the adjustment effectively. A somewhat related question: "What problem or game are variance and standard deviation optimal solutions for? " with a great answer from @whuber. $\endgroup$
    – Richard Hardy
    Commented Jan 7, 2020 at 9:03
  • $\begingroup$ Could you please clarify "absolute loss"? One reason for asking is that the natural measures of loss in problems related to variance will be functions of their ratios, not their differences. $\endgroup$
    – whuber ♦
    Commented Jan 10, 2020 at 14:51
  • $\begingroup$ @whuber, I have edited the question. My understanding of the matter is incomplete, so thank you for your help in formulating the question in a clearer and/or more sensible way. $\endgroup$
    – Richard Hardy
    Commented Jan 10, 2020 at 15:29

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.