 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2008/iii/paper-46/3/b/i/solution

Codex  0 Created 2026-10-03 Updated 2026-10-14

Put

d = \sum_{i} v_{i}

. With common Weibull distribution parameters and the hazard from part (a), substitution into the general log-likelihood gives

ℓ (λ, p) = d lo g p + p d lo g λ + (p - 1) i \sum v_{i} lo g x_{i} - λ^{p} i \sum x_{i}^{p} .

This is up to censoring-mechanism terms independent of

λ, p

. For

d > 0

and fixed

p

, its maximizing rate is

(d / \sum_{i} x_{i}^{p})^{1/ p}

; the shape still requires separate estimation if it is unknown.

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