As $b\downarrow0$, the Huber estimating equation approaches the sign equation
$$
\sum_i\operatorname{sgn}(x_i-\theta)=0,
$$
whose solution is the <sample median>. Hence the most B-robust location M-estimator is the median, with minimum <gross-error sensitivity>
$$
\gamma^*=\frac1{2\phi(0)}=\sqrt{\frac\pi2}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 223 / 2 / c / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-223/2/c/solution