The transformation $g(t)=e^{t/2}$ satisfies $g(\theta)=\sigma$ and $g'(\theta)=\sigma/2$. Applying the <delta method> to part c gives
$$
\sqrt n\left(e^{T_n/2}-\sigma\right)
\xrightarrow d
N\!\left(0,
\frac{\sigma^2}{4}
\frac{\mathbb E_{G_0}[\psi(W)^2]}
{\{\mathbb E_{G_0}[\psi'(W)]\}^2}
\right).
$$
Thus exponentiating half the robust location estimate produces an asymptotically normal <scale estimator>.

Solution

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

A scale estimator measures the spread of a distribution and transforms proportionally when all observations are multiplied by a positive constant.


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