mnb96
Nov4-10, 06:53 PM
Hello,
we are given N independent random variables z_i defined as follows:
z_i = \theta + v_i
where the r.v. v_i are zero-mean normal distributions v_i \sim N(0,\sigma^2).
I want to compute the variance of the estimator
\hat{\theta}=\frac{1}{n}\sum_{i=1}^n z_i
However I can't seem to get the right result (I get always zero).
I computed first E(\hat{\theta})^2=\theta^2.
Then I try to compute E(\hat{\theta^2}) as follows:
\frac{1}{n^2}E\left[\left(\sum_{i=1}^n z_i \right)^2\right] = \frac{1}{n^2}E\left[\left(\sum_{i=1}^n (z_i)^2 \right) + \left(\sum_{i=1}^n z_i \right) \left( \sum_{j\neq i} z_j \right) \right]
= \frac{1}{n^2}E\left[\sum_{i=1}^n z_i^2 \right] + \frac{n-1}{n}\theta^2
= \frac{1}{n^2}E\left[\sum_{i=1}^n (\theta^2 +2\theta v_i + v_i^2) \right] + \frac{n-1}{n}\theta^2
= \frac{1}{n}\theta^2 + \frac{n-1}{n}\theta^2
From this we get Var(\hat{\theta})=E(\hat{\theta}^2)-E(\hat{\theta})^2 = 0.
Where is the mistake?
we are given N independent random variables z_i defined as follows:
z_i = \theta + v_i
where the r.v. v_i are zero-mean normal distributions v_i \sim N(0,\sigma^2).
I want to compute the variance of the estimator
\hat{\theta}=\frac{1}{n}\sum_{i=1}^n z_i
However I can't seem to get the right result (I get always zero).
I computed first E(\hat{\theta})^2=\theta^2.
Then I try to compute E(\hat{\theta^2}) as follows:
\frac{1}{n^2}E\left[\left(\sum_{i=1}^n z_i \right)^2\right] = \frac{1}{n^2}E\left[\left(\sum_{i=1}^n (z_i)^2 \right) + \left(\sum_{i=1}^n z_i \right) \left( \sum_{j\neq i} z_j \right) \right]
= \frac{1}{n^2}E\left[\sum_{i=1}^n z_i^2 \right] + \frac{n-1}{n}\theta^2
= \frac{1}{n^2}E\left[\sum_{i=1}^n (\theta^2 +2\theta v_i + v_i^2) \right] + \frac{n-1}{n}\theta^2
= \frac{1}{n}\theta^2 + \frac{n-1}{n}\theta^2
From this we get Var(\hat{\theta})=E(\hat{\theta}^2)-E(\hat{\theta})^2 = 0.
Where is the mistake?