M"Unbiased Estimator for Sigma: Theorem 3.3.1 and Practical Application | CCM

[happyg1]

Hi, I'm working on the following problem and I need some clarification:
Suppose that a sample is drawn from a N(\mu,\sigma^2) distribution. Recall that \frac{(n-1)S^2}{\sigma^2} has a \chi^2 distribution. Use theorem 3.3.1 to determine an unbiased estimator of \sigma
Thoerem 3.3.1 states:
Let X have a \chi^2(r) distribution. If k>-\frac{r}{2} then E(X^k) exists and is given by:
E(X^k)=\frac{2^k(\Gamma(\frac{r}{2}+k))}{\Gamma(\frac{r}{2})}
My understanding is this:
The unbiased estimator equals exactly what it's estimating, so E(\frac{(n-1)S^2}{\sigma^2})is supposed to be\sigma^2 which is 2(n-1).
Am I going the right way here?
CC

 

[happyg1]

Ok, So after hours of staring at this thing, here's what I did:
I let k=1/2 and r=n-1, so the thing looks like this:
E<s>=\sigma(\sqrt{\frac{2}{n-1}}\frac{\Gamma\frac{n}{2}}{\Gamma\frac{n-1}{2}}</s>
so I use the property of the gamma function that says:
\Gamma(\alpha)=(\alpha-1)!
which leads to:
E<s>=\sigma\sqrt\frac{2}{n-1}(n-1)</s>
So now do i just flip over everything on the RHS,leaving \sigma by itself and that's the unbiased estimator, i.e.
\sqrt{2(n-1)}E<s>=\sigma</s>
Any input will be appreciated.
CC

 

[happyg1]

OK
Anyone who looked and ran away, here at last is the solution: (finally)
E<s>=\sigma\sqrt{\frac{2}{n-1}} \frac{\Gamma\frac({n}{2})}{\Gamma\frac({n-1}{2})}</s>
is indeed correct, however my attempt to reduce the RHS with the properties of the Gamma function is wrong.
The unbiased estimator is obtained by isolating the \sigma on the RHS and then using properties of the Expectation to get:
E\left(\sqrt\frac{n-1}{2}\frac{\Gamma(\frac{n-1}{2})}{\frac\Gamma(\frac{n}{2})}S\right)=\sigma
So at last it has been resolved. WWWWEEEEEEEEEEEeeeeeeee
CC