Let $\theta >0$ be a parameter and let $X_1,X_2,\ldots,X_n$ be a random sample with pdf $f(x\mid\theta)=\frac{1}{3\theta}$ if $-\theta \leq x\leq 2\theta$ and $0$ otherwise.

a) Find the MLE of $\theta$

b) Is the MLE a sufficient statistic for $\theta$?

c) Is the MLE a complete statistic for $\theta$?

d) Is $\frac{n+1}{n}\cdot MLE$ the UMVUE of $\theta$?

I've been able to solve a). The MLE of $\theta$ is $\max(-X_{(1)},\frac{X_{(n)}}{2}).$ Also, you can show that it is sufficient using the Factorization Theorem.

However, I cannot solve the next questions I think because of the $\max$ in the MLE. Is there another way to express $\max(-X_{(1)},\frac{X_{(n)}}{2})$? Can I express the MLE as $\frac{|X|_{(n)}}{2}?$