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

Table of contents
- Solution d

Solution

The transformation

g (t) = e^{t /2}

satisfies

g (θ) = σ

and

g^{'} (θ) = σ /2

. Applying the delta method to part c gives

n (e^{T_{n} /2} - σ) d N (0, \frac{σ ^{2}}{4} \frac{E _{G_{0}} [ ψ ( W ) ^{2} ]}{{ E _{G_{0}} [ ψ ^{'} ( W )] } ^{2}}) .

Thus exponentiating half the robust location estimate produces an asymptotically normal scale estimator.

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