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

So, I set up $\displaystyle \theta + \theta = \theta^r$

$\displaystyle 2 \theta = \theta^r$

$\displaystyle log 2 + log \theta = r log \theta$

$\displaystyle r = 1 + log 2 / log \theta$

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

Any help would be appreciated. Thanks.