Let $X_1, ..., X_n$ be from a sample from a probability distribution:
$$p_{\theta}(x) = \frac{2x}{\theta^2}1_{0{\leq}x{\leq}\theta}$$
, where $\theta > 0$ is an unknown parameter.
I have found from (i) and (ii) of the question that the unbiased method of moments estimator for $\theta$ is $\frac{3\bar{X}}{2}$.
(iii) asks for an estimator for $\theta^2$, which I found to be $2\bar{X^2}$ using the second moment of the distribution. However, this is inconsistent with the answer in my textbook which says that it is $(\frac{3\bar{x}}{2})^2$. I am confused as to the method my textbook used to arrive at this answer. It seems that the estimator for $\theta$ is simply squared, which I assumed is only possible for Maximum Likelihood Estimators when the function of $\theta$ is bijective.
(iv) asks for an unbiased estimator for $\theta^2$, which if I were to use the estimator as given by my textbook, I would require the variance of the distribution to compute, which I am unable to obtain.
Would really appreciate some help on this. Thanks!