# Variance of S^2

• Nov 27th 2012, 02:22 PM
Scopur
Variance of S^2
Haven't used this for a while.. not sure if i get the texing.
I prove first that (this was fine)
$\displaystyle \frac{1}{2n(n-1)} \sum^n_{i=1}\sum^n_{j=1}(X_i-X_j)^2 = S^2$

Then i am to calculate, assuming all the $\displaystyle X_i's$ have finite fourth moment, denoting $\displaystyle \theta_1 =EX_i, \theta_j=E(X_i - \theta_1)^j, j=1,2,3$, the
$\displaystyle Var(S^2)=\frac{1}{n}\left(\theta_4-\frac{n-3}{n-1}\theta_2^2 \right)$. I first used $\displaystyle Var(S^2)=E(S^4)-E(S^2)^2$ and tried to calculate $\displaystyle E(S^2)$ but
I don't really get how these cross terms work. My first attempt i got$\displaystyle \theta_2/(n-1)$ but i'm only assuming that's wrong.
• Nov 27th 2012, 07:10 PM
Scopur
Re: Variance of S^2
$\displaystyle Var (S^2) = E(S^4)-E(S^2)^2$
We know that$\displaystyle E(S^2) = \theta_1^2$ so we just need to calculate $\displaystyle E(S^4)$. Let $\displaystyle Y_i = X_i - \theta_2$, for $\displaystyle i=1,2,...n$. Then put
$\displaystyle S^2 =\frac{n \sum_{i=1}^n Y_i^2-(\sum_{i=1}^n Y_i)^2}{n(n-1)}$
So that
$\displaystyle S^4 = \frac{n^2(\sum Y_i^2)^2 -2n(\sum Y_i^2)(\sum Y_i)^2 +(\sum Y_i)^4}{n^2(n-1)^2}$
and
$\displaystyle E(S^4) = \frac{n^2E(\sum Y_i^2)^2 -2nE[(\sum Y_i^2)(\sum Y_i)^2] +E(\sum Y_i)^4}{n^2(n-1)^2}$

Now we have $\displaystyle i \neq j \neq k$
$\displaystyle E(Y_i Y_j) = 0, E(Y_i^3Y_j)=0, E(Y_i^2Y_jY_k)=0, E(Y_i^2Y_j^2) = \theta_2, E(Y_i^4)= \theta_4$
and
$\displaystyle E(\sum Y_i^2)^2 = n\theta_4 + n(n-1)\theta_2^2 E[ (\sum Y_i^2)(\sum Y_i)^2] = n\theta_4 + 3n(n-1)\theta_2^2 E(\sum Y_i)^4 = n\theta_4 + 3n(n-1) \theta_2^2$
Giving us
$\displaystyle E(S^4) = \frac{n^2(n\theta_4 + n(n-1)\theta_2^2) -2n(n\theta_4 + 3n(n-1)\theta_2^2) + n\theta_4 + 3n(n-1) \theta_2^2}{n^2(n-1)} = \frac{(n-1)\theta_4 +(n^2 -2n +3)\theta_2^2}{n(n-1)}$
Which gives
$\displaystyle Var(S^2) = \frac{(n-1)\theta_4 +(n^2 -2n +3)\theta_2^2}{n(n-1)} - \theta_2^2 \\ = \frac{1}{n} \left (\theta_4 -\frac{n-3}{n-1} \theta_2^2 \right)$

Not sure why it hates my teXing... anyways its solved