# When is it better to have an unbiased estimator instead of one that has a smaller risk?

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

• 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?
– Dave
Commented Oct 25, 2022 at 3:02
• @Dave For n-1 at least, it's unbiased but it may not be UMVUE I'd think.
– Phil
Commented Oct 25, 2022 at 3:33
• 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.] Commented Oct 25, 2022 at 4:10
• @angryvavian interesting thank you. Maybe squared error loss isn't the best loss function for variance then hmm
– Phil
Commented Oct 25, 2022 at 12:55