 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/ii/paper-2/29l/d/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

The posterior is Pareto with shape

α + n

and lower endpoint

m = max {ℓ, X_{(n)}}

. Minimizing

E [(δ - θ)^{2} / θ^{2} ∣ X]

gives

δ = E (θ^{- 1} ∣ X) / E (θ^{- 2} ∣ X)

, hence

\hat{θ}_{α, ℓ} = m \frac{α + n + 2}{α + n + 1} .

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