Solution
= Solution
Neither proper gamma prior yields the condition in part (c). The risk in part (a) is unbounded as $θ\to∞$ because $β>0$. The risk in part (b) is analytic, tends to zero at infinity, and is not identically zero, so its prior average is strictly below its supremum for a gamma prior with positive density everywhere. Thus this sufficient condition proves neither estimator minimax for any $α,β>0$.