I am trying to make $\hat \theta$ an unbiased estimator for a uniform distribution on $(0, \theta)$. So I am given the fact that $\hat \theta$ is the MLE for $\theta$, but I need to show that if I choose $c_1$, then the estimator becomes unbiased. What gets me stuck is the fact that $\hat \theta = \max(X_1, ..., X_n)$.
Basically, how do I deal with the "max" part of the problem given that I want to show that $\hat \theta$ can be unbiased if we choose $c_1$ such that $E[\hat\theta]$ = $\theta$.
ie. I need to find $c_1$ such that $\hat \theta = c_1 \max(X_1, ..., X_n)$ such that $\hat \theta$ is unbiased (aka $E(\hat \theta) = \theta)$
Thanks!
