Let $X_1, ..., X_n$ identical independent variables that follows a beta distribution $\text{Beta}(\theta,1)$
Is $S = \sum_{i=1}^{n}X_i$ a complete statistic for $\theta$?
I have proved that the sum is not a sufficient statistic for $\theta$ but I'm having a difficult time about completeness.
If the statistic is not complete, then I need to find a function $g$ such that $E[g(S)] = 0 $ implies $g(S) = 0$ almost surely for every $\theta$. Unfortunately I have not been able to find such function.
I would really appreciate any hints or sugestions with this problem