Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 223 / 1 / a / Solution

Put $U=X/\sigma$. The standardized efficient score function for scale isAn optimal B-robust scale M-estimator solveswhere the optimal bounded score has the clipped-score formThe constants $A,z$ enforce Fisher consistency, the chosen normalization, and the clipping bound $b$ on the influence function.

For the standard normal distribution, $f^{\prime }(u)/f(u)=-u$, hence $\Lambda (u)=u^2-1$. The score therefore simplifies towith $z$ chosen so that $\mathbb{E}\psi (Z)=0$ for $Z\sim N(0,1)$.