The question is: Let X_1, X_2 be iid samples from a Bin(1, \theta) and assume that we want to estimate \gamma(\theta) = \theta^r, r \in \mathbb{R}. Find the possible range of r such that \gamma(\theta) is U-estimable based on T(X) = X_1 + X_2.

So, I set up \theta + \theta = \theta^r
2 \theta = \theta^r
log 2 + log \theta = r log \theta
r = 1 + log 2 / log \theta

However, if \theta = .5, we have r = 0, but that doesn't satisfy 2 \theta = \theta^r.

Any help would be appreciated. Thanks.