- #1
mnb96
- 715
- 5
Hello,
we are given N independent random variables [itex]z_i[/itex] defined as follows:
[tex]z_i = \theta + v_i[/tex]
where the r.v. [itex]v_i[/itex] are zero-mean normal distributions [itex]v_i \sim N(0,\sigma^2)[/itex].
I want to compute the variance of the estimator
[tex]\hat{\theta}=\frac{1}{n}\sum_{i=1}^n z_i[/tex]
However I can't seem to get the right result (I get always zero).
I computed first [itex]E(\hat{\theta})^2=\theta^2[/itex].
Then I try to compute [itex]E(\hat{\theta^2})[/itex] as follows:
[tex]\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][/tex]
[tex]= \frac{1}{n^2}E\left[\sum_{i=1}^n z_i^2 \right] + \frac{n-1}{n}\theta^2[/tex]
[tex]= \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[/tex]
[tex]= \frac{1}{n}\theta^2 + \frac{n-1}{n}\theta^2[/tex]
From this we get [tex]Var(\hat{\theta})=E(\hat{\theta}^2)-E(\hat{\theta})^2 = 0[/tex].
Where is the mistake?
we are given N independent random variables [itex]z_i[/itex] defined as follows:
[tex]z_i = \theta + v_i[/tex]
where the r.v. [itex]v_i[/itex] are zero-mean normal distributions [itex]v_i \sim N(0,\sigma^2)[/itex].
I want to compute the variance of the estimator
[tex]\hat{\theta}=\frac{1}{n}\sum_{i=1}^n z_i[/tex]
However I can't seem to get the right result (I get always zero).
I computed first [itex]E(\hat{\theta})^2=\theta^2[/itex].
Then I try to compute [itex]E(\hat{\theta^2})[/itex] as follows:
[tex]\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][/tex]
[tex]= \frac{1}{n^2}E\left[\sum_{i=1}^n z_i^2 \right] + \frac{n-1}{n}\theta^2[/tex]
[tex]= \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[/tex]
[tex]= \frac{1}{n}\theta^2 + \frac{n-1}{n}\theta^2[/tex]
From this we get [tex]Var(\hat{\theta})=E(\hat{\theta}^2)-E(\hat{\theta})^2 = 0[/tex].
Where is the mistake?