Is the method of moments estimator for θ unbiased?

[songoku]

Homework Statement

Please see below

Relevant Equations

Method of Moments Estimator (MME)

(i)
$$E(X)=\bar X$$
$$(-1)\left(\frac{\theta}{2}\right)+(1)\left(\frac{\theta}{2}\right)=\bar X$$
$$\bar X=0$$

Then:
$$\text{Var} (X)=\bar {X^2}-(\bar X)^2$$
$$(1)\left(\frac{\theta}{2}\right)+(1)\left(\frac{\theta}{2}\right)=\bar {X^2} - 0$$
$$\theta = \frac{1}{n} \sum_{i=1}^{n} X_{i}^{2}$$

(ii)
$$E(\hat {\theta})=E(\bar {X^2})=\theta$$

So the MME is unbiased estimator.

Is my working correct? Thanks

 

[Gavran]

(i) Your work is correct, but your notation should be clearer.

$$\theta = \frac{1}{n} \sum_{i=1}^{n} X_{i}^{2}$$

should be $$ \hat{\theta}=\frac1n\sum_{i=1}^{n}X_i^2 $$.
(ii) I do not understand how you get this

$$E(\hat {\theta})=E(\bar {X^2})=\theta$$

.
By following steps
$$ E(\hat{\theta})=E(\frac1n\sum_{i=1}^{n}X_i^2)=\frac1nE(\sum_{i=1}^{n}X_i^2)=... $$
you should get this $$ E(\hat{\theta})=\bar{X^2}=\theta $$ at the end.

 

[songoku]

I understand.

Thank you very much Gavran