Let $Y_1,Y_2,...,Y_n$ be a random sample from $N(\mu,\sigma^2)$.
I need to show that $S$ is a biased estimator of $\sigma$.
As from the definition, I see that $\frac{(n-1)S^2}{\sigma^2}\sim\chi^{2}_{(n-1)}$.
My thought process follows like this:
- If $X\sim \chi^2_{n-1}$, then by definition $X\sim Gamma(\frac{n-1}{2},2)$.
- Apply above definition to this problem. Since I need to find if $S$ is biased or not, I must compute the expected value of S.
- I also know that $E[Y^a]=\frac{\beta^a\Gamma(\alpha+a)}{\Gamma({\alpha})}$ when Y is a gamma distribution. I should be able to apply the same concept to this problem.
After calculations, I obtain something like this:
$E[S]=\frac{\sqrt{2}\Gamma(\frac{n}{2})}{\Gamma{(\frac{n-1}{2})}}\times\frac{\sigma}{n-1}$
How do I prove that above is an biased estimator of $\sigma$? Obviously the value of the gamma functions change as the n changes (between even and odd). Is there a more systematic way of obtaining this? I'm not quite sure how to deal with this gamma function and go on to compare $E[S]$ with $\sigma$. Is there a way for me to simplify this gamma function further?