# Uniform - unbiased estimators?

• Apr 17th 2010, 08:16 PM
Statistik
Uniform - unbiased estimators?
Hi,

I have an iid sample from $U(\theta_1, \theta_2)$ and found the MM of $(\theta_1, \theta_2)$ to be:

$
\theta_1 = \bar X_n - \sqrt {3 Sn^2}$
and $\theta_2 = \bar X_n + \sqrt {3 Sn^2}
$

In our solutions, we are told to notice that

(i) $\frac {1} {n} \sum_i {(X_i)^2} - (\frac {1} {n} \sum_i {X_i})^2 = \frac {n-1} {n} {S_n}^2
$

and

(ii) $E(3(\frac {1} {n} \sum_i {(X_i)^2} - (\frac {1} {n} \sum_i {X_i})^2)) = \frac {n-1} {n} \frac {(\theta_2 - \theta_1)^2} {4}
$

I'd love to have help on how to actually see this! Thank you!