1
$\begingroup$

Let $X_1,X_2$,....,be a random sample from $N(q,w^2)$; $q,w$ are unknown. Let $S_n$ be the sample standard deviation.

i.e $S_n^2=\frac{1}{n-1}\sum(X_i-\bar{X})^2$

What is $Var(S_n^2)$? and how to show that $S_n^2$ is asymptotically normal?

I tried to do Variance part using moment generating functions but expressions are getting extremely complicated.

$\endgroup$
  • $\begingroup$ Welcome to the site. I've improved your formatting but you should check that I haven't introduced any errors. If you want your posts to be readable you can use latex code in your questions, inserted between $ signs. $\endgroup$ – Peter Ellis Apr 22 '12 at 19:41
  • $\begingroup$ Is this homework? If so, you should add the "homework" tag. $\endgroup$ – Peter Ellis Apr 22 '12 at 19:41
  • $\begingroup$ Do you know the finite-sample distribution of $S_n^2$? If so, you can start from there, and see what happens as $n \to \infty$. $\endgroup$ – jbowman Apr 22 '12 at 19:48
  • $\begingroup$ I donot know distribution of $S_n^2$. $\endgroup$ – Questions-Math Apr 22 '12 at 20:05
  • $\begingroup$ Also I need to show that $\sqrt(n)(S_n^2-w^2)->N(0,m)$ for some m function of w as n->$\infty$ in distribution.That is the meaning of asymptotic normality. $\endgroup$ – Questions-Math Apr 22 '12 at 20:11
1
$\begingroup$

I got it! I came to know that $(n-1)S_n^2/w^2$ is distributed as $\chi^2_{n-1}$. From there I can calculate the variance.

Now $S_n^2/w^2$ is the sum of $n-1$ $\chi^2_1$ random variables divided by $(n-1)$.

From central limit theorem, $\sqrt{n-1}(S_n^2/w^2 - 1)\xrightarrow{D} N(0,2)$. Hence the asymptotic normality.

$\endgroup$
  • 1
    $\begingroup$ I've cleaned up the language. The general idea is ok, but the answer needs some polish. $\endgroup$ – mpiktas Apr 23 '12 at 9:16

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.