Watching a video by the "mathematicalmonk" on the web, I was wondering how to answer this kind of questions:

Given $X_1,\ldots,X_n\sim \mathcal{N}\left(\mu,\sigma^2\right)$. Assume that $\mu$ is unknown. Using the square loss $\mathcal{L}\left(a,b\right)=\left(a-b\right)^2$:

1) Does $\hat{\sigma}^2=\frac{1}{n}\sum_{i=1}^n(X_i-\bar{X})^2$ dominate* $s^2=\frac{n}{n-1}\hat{\sigma}^2$? Or viceversa?

2) Are $\hat{\sigma}^2$ and $s^2$ admissible**?

3) Is there a "best" $s_c^2=c\hat{\sigma}^2$ for some $c\geq 0$?

*given decision rules $a,b$, $a$ dominates $b$ if $\mathcal{R}(\theta,a)\leq\mathcal{R}(\theta,b),\,\, \forall \theta \in\Theta$ and $\mathcal{R}(\theta,a)<\mathcal{R}(\theta,b),\,\,$ for some $\theta \in\Theta$, where $\mathcal{R}(\theta,f)=\mathbb{E}\left[\mathcal{L}\left(\theta,f\left(D\right)\right)|\theta\right]$, and $D$ is the sample.

**A decision rule $a$ is inadmissible if there is another decision rule which dominates $a$. Otherwise, $a$ is admissible.

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    $\begingroup$ Since this looks like routine book-work it should probably carry the self-study tag (and you should read the tag wiki info at the link for how such questions would generally work). There's at least one tag on your question that doesn't seem to be needed, so there seems ot be room for the new tag to go in its place. $\endgroup$ – Glen_b Mar 5 '14 at 1:42