Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 223 / 3 / b

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- Solution b

Solution

The functional is

T (F) = \int ψ (x) d F (x), ψ (x) = Δ (x - \frac{θ _{0} + θ _{1}}{2}) .

The influence function of an expectation functional is

IF_{test} (x; T, F_{θ}) = ψ (x) - E_{F_{θ}} ψ (X) .

Because

ψ

is affine with nonzero slope, this is unbounded under both

F_{θ_{0}}

and

F_{θ_{1}}

.

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