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

Let $m=F^{-1}(1/2)$. Since $F$ has an everywhere positive density, it is continuous and strictly increasing, so $F(m)=1/2$. Membership in the Kolmogorov neighborhood of a distribution givesBy the symmetry of the standard normal distribution,The stated asymptotic-bias formula for the sample median therefore yields

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.

DefineThus $F_{+}$ is the equal mixture of $N(0,1)$ and $N(2b_1,1)$, while $F_{-}$ is the equal mixture of $N(0,1)$ and $N(-2b_1,1)$. They have finite variance and everywhere positive densities. Normal symmetry shows that $F_{+}$ is symmetric about $b_1$ and $F_{-}$ about $-b_1$.

For every $t$,where the maximum occurs at $t=b_1$. ConsequentlyThe same argument, shifted and reflected, applies to $F_{-}$. Hence both distributions belong to ${\mathcal{P}}_{\epsilon }^K(\Phi )\cap \mathcal{M}$ and satisfy the required translation relation.