Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 223 / 1 / b / Solution

WriteIf $X_{+}\sim F_{+}$, then $X_{+}-2b_1\sim F_{-}$ because $F_{-}(t)=F_{+}(t+2b_1)$. The translation-invariant estimator property givesFor every real $u$,Taking $u={\mu }_{+}$ shows that every admissible estimator has maximum asymptotic bias at least $b_1$ on the pair $\{F_{+},F_{-}\}$. ThereforeTogether with part a, this proves the minimax asymptotic bias optimality of the median.