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

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 $Z\sim N(0,1)$.

Write $X=\sigma Z$ with $Z\sim F_1$. Thenwhere the distribution of $W=\log Z^2$ does not depend on $\sigma$. Consequentlywhich is a location family. Since $\sigma >0$, the transformed parameter $\theta =\log {\sigma }^2$ ranges over all of $\mathbb{R}$; the paper's restriction $\theta >0$ is unnecessary.

Let $W=Y-\theta \sim G_0$ and suppose $a={\mathbb{E}}_{G_0}[{\psi }^{\prime }(W)]\ne 0$. A first-order expansion of the estimating equation $n^{-1}{\sum }_i\psi (Y_i-T_n)=0$ gives the asymptotic linear representationThe central limit theorem therefore yields

The transformation $g(t)=e^{t/2}$ satisfies $g(\theta )=\sigma$ and $g^{\prime }(\theta )=\sigma /2$. Applying the delta method to part c givesThus exponentiating half the robust location estimate produces an asymptotically normal scale estimator.