MoM of Uniform Distribution

Question

I'm trying to learn MoM. I'd like to find MoM and show $\hat z_{MoM} $ is unbiased for $U(- \sqrt z, \sqrt z)$, which has a density function:

$f(x,z)= \begin{cases} \frac{1}{2\sqrt z} & -\sqrt z \leq x \leq \sqrt z \\ 0 & otherwise \end{cases} $

$M_1=\int^{\sqrt z}_{- \sqrt z}\frac{x}{2 \sqrt z}dx=0=\frac{1}{n}\sum^{n}_{i=1}X_i$

$M_2=\int^{\sqrt z}_{- \sqrt z}\frac{x^2}{2 \sqrt z}dx=\frac{z}{3}=\frac{1}{n}\sum^{n}_{i=1}X^2_i$

$E[x^2]=\bar x=\frac{z}{3}$

From these, I can say $\hat z_2 =3 \bar x$, since the first moment give me nothing I can take the second one right?

For showing the unbiased part:

$E[\hat z_2]=E \Bigg [ \Bigg(\frac{1}{n}\sum^{n}_{i=1}X^2_i \Bigg )^2 \Bigg ]=\frac{1}{9}\sum^{n}_{i=1}\sum^{n}_{j=1}E[X^2_iX^2_j]$, I couldn't go any further.

Any help would be appreciated!

Answer 1

Note that we do not have $\bar{x} = \frac{z}3$. From your working,

$$\hat{z}_{MOM}= \frac{3}n\sum_{i=1}^n X_i^2$$

\begin{align} E[\hat{z}_{MON}]=\frac3n \sum_{i=1}^n E[X_i^2]=\frac3n\cdot n\left( \frac{z}3\right)=z \end{align}