Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 4 / 28J / c / Solution

An admissible Bayes estimator whose Bayes risk equals its maximum risk is minimax. Indeed every estimator has maximum risk at least its prior-average risk, which is at least the Bayes risk; equality for the stated estimator attains this lower bound. Admissibility excludes a distinct estimator improving it everywhere.