Write
$$
\mu_+=\lim_{n\to\infty}\mathbb E_{F_+}T_n.
$$
If $X_+\sim F_+$, then $X_+-2b_1\sim F_-$ because $F_-(t)=F_+(t+2b_1)$. The <translation-invariant estimator> property gives
$$
\mu_-:=\lim_{n\to\infty}\mathbb E_{F_-}T_n
=\mu_+-2b_1.
$$
For every real $u$,
$$
\max\{|u|,|u-2b_1|\}\geq b_1.
$$
Taking $u=\mu_+$ shows that every admissible estimator has maximum asymptotic bias at least $b_1$ on the pair $\{F_+,F_-\}$. Therefore
$$
\inf_{\{T_n\}\subset\mathcal T}
\sup_{F\in\mathcal P_\varepsilon^K(\Phi)\cap\mathcal M}
b(\{T_n\},F)\geq b_1.
$$
Together with part a, this proves the <minimax asymptotic bias> optimality of the median.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-223/1/b/solution