I am having an issue finding a best unbiased estimator for $\theta$. Any help is appreciated.
Let $X_1, ..., X_n$ be a random sample from a population with pdf: $f(x\mid\theta)=\frac{1}{2\theta}$ $-\theta<x<\theta,\, \theta>0$.
I understand that $T(x) := |X_n|$ is the sufficient statistic since by the factorization theorem we have:
$$f(x\mid \theta) = \left(\frac{1}{2\theta}\right)^n \prod_{i=1}^n I[|x_i|<\theta]$$
I think my main issue is showing that this sufficient statistic is also a complete sufficient statistic. Can somebody please aid me in this?