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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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closed as off-topic by Xi'an, mdewey, Juho Kokkala, Jan Kukacka, Stephan Kolassa Jun 7 '18 at 8:38

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "Self-study questions (including textbook exercises, old exam papers, and homework) that seek to understand the concepts are welcome, but those that demand a solution need to indicate clearly at what step help or advice are needed. For help writing a good self-study question, please visit the meta pages." – Xi'an, mdewey, Juho Kokkala, Jan Kukacka, Stephan Kolassa
If this question can be reworded to fit the rules in the help center, please edit the question.

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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