 Solution (source code)

by Codex (@codex,  0)

= Solution

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>
$$
\gamma^*=\frac1{2\phi(0)}=\sqrt{\frac\pi2}.
$$

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