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I just learned that for $X_1, \ldots X_n \sim N(\mu, \sigma^2)$ i.i.d, the sample variance $\frac{1}{n-1} \sum_{i=1}^n (X_i - \bar X)^2$ is unbiased, and it is in fact UMVUE.

However, it is not admissible, since $\frac{1}{n+1} \sum_{i=1}^n (X_i - \bar X)^2$ minimizes the squared error risk.

So why is it that we use $\frac{1}{n-1}$ instead of $\frac{1}{n+1}$? In general, when is it better to have an unbiased estimator, even though it may not be admissible?

Thank you

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  • $\begingroup$ Interesting question and a +1. Some thoughts of mine: What happens to the $n+1$ estimator if your distribution is not Gaussian? What about the $n-1$ estimator? $\endgroup$
    – Dave
    Oct 25, 2022 at 3:02
  • $\begingroup$ @Dave For n-1 at least, it's unbiased but it may not be UMVUE I'd think. $\endgroup$
    – Phil
    Oct 25, 2022 at 3:33
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    $\begingroup$ This answer quotes Stein: "I find it hard to take the problem of estimating a [variance] with quadratic loss function very seriously." Perhaps it is something like this: to talk about admissibility, one needs to agree on a loss function, whereas one can discuss unbiasedness without a loss function, which might be why people have traditionally favored the unbiased estimator? [I am speculating here.] $\endgroup$
    – angryavian
    Oct 25, 2022 at 4:10
  • $\begingroup$ @angryvavian interesting thank you. Maybe squared error loss isn't the best loss function for variance then hmm $\endgroup$
    – Phil
    Oct 25, 2022 at 12:55

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