Assume $X_i$, where $i = 1,...,n$ are random samples from $Unif(-\theta, \theta)$. Find a method of moments estimator $T(x)$ of $\theta$ and check it is biased or not.
I have already derived that $\hat \theta_2 = \sqrt{\frac{3}{n} \sum_{i=1}^{n} X_i^2}$, and I know I should prove $E[\hat \theta_2] = \theta$, but I don't know how to write the process of the calculation. Could anyone give me a hint or help? Thanks
By Jensen's inequality, for $\theta>0$, $E[\hat\theta_2]=E\left[\sqrt{\frac{3}{n} \sum_{i=1}^{n} X_i^2}\right]< \sqrt{E\left[\frac{3}{n} \sum_{i=1}^{n} X_i^2\right]}=\sqrt{\frac{3}{n} \sum_{i=1}^{n} \text{Var}(X_i)}=\sqrt{\frac{3}{n} \sum_{i=1}^{n} \frac{\theta^2}{3}}=\theta$