Let $X_1, X_2, \dots, X_n$ be a sample of i.i.d. random variables, with density $f_\theta=\frac{2}{3\theta}\left(1-\frac{x}{3\theta}\right) $ for $0 < x < 3\theta$. And $f_\theta=0$ if $ x < 0$ or $ x>3\theta$
Let $\hat{\theta}=\overline{X}$ be an estimate for $\theta$
1.) Determine whether $\theta$ is unbiased?
2.) Determine whether $\theta$ is consistent?
3.) Determine whether $\theta$ is sufficient?
4.) Why doesn't the Cramer-Rao lower bound apply to unbiased estimates of $\theta$ for this distribution?
I tried:
1.) $\theta$ is unbiased because the integral of $\int_0^{3\theta}{x}\left(\frac{2}{3\theta}\left(1-\frac{x}{3\theta}\right)\right) \, dx = \theta = E[\hat{\theta}]$Hence the statistic is unbiased.
2.) Yes, because $\operatorname{VAR}[\hat{\theta}]=\operatorname{VAR}\left[\overline{X}\right]=\frac{\sigma^2}{n}. \lim_{n\to \infty}\left(\frac{\sigma^2}{n}\right)=0$Hence $\theta$ is consistent.
3.)According to the factorization theorem I have to find a function $g(\hat{\theta},\theta)$ and $h(x_1,x_2,\dots ,x_n)$ so that $gh=f(x_1,x_2,...,x_n; \theta)$
I guess I have to calculate $\prod_{i=1}^{n}{f_\theta} $ and derive some function g. But I don't know how to start. Thank you for your help in advance!