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

The normal location score is $x-\theta$. Under a bound on gross-error sensitivity, the variance-minimizing influence curve clips this score. For $b>0$, define the Huber scoreThe optimal B-robust estimator is the Huber location estimator ${\hat{\theta }}_b$ defined byEquivalently, it has the explicit optimization formwhere the Huber loss isAt $N(\theta ,1)$ its normalized influence function is

The gross-error bound corresponding to $b$ isIt is strictly increasing becauseso the numerator of $c^{\prime }(b)$ is positive. FurthermoreThus $b\in (0,\infty )$ corresponds exactly to $c\in (\sqrt{\pi /2},\infty )$.

As $b\downarrow 0$, the Huber estimating equation approaches the sign equationwhose solution is the sample median. Hence the most B-robust location M-estimator is the median, with minimum gross-error sensitivity