Solution
= Solution
Put $r=1-\alpha\in(0,1)$. The variable $Y=X_1-\theta$ has density $ry^{r-1}$ on $(0,1)$, so $\mathbb EY^j=r/(r+j)$. Thus
$$
\boxed{\mathbb EX_1=\theta+\frac r{r+1},\qquad
\operatorname{Var}(X_1)=\frac r{(r+2)(r+1)^2}.}
$$
The <unbiased estimator> is
$$
\boxed{\widetilde\theta_n=\overline X_n-\frac{1-\alpha}{2-\alpha},\qquad c(\alpha)=-\frac{1-\alpha}{2-\alpha}.}
$$
Taking <expectations> verifies that its <bias> is zero.