 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2001/iii/paper-29/2/b/i/solution

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

Assume independent individuals and independent censoring conditional on their explanatory variables, with the censoring mechanism carrying no parameter

θ

. A failure contributes its failure-time probability density function

f_{i} (x_{i}; θ)

, while a censored observation contributes the survival function

F_{i} (x_{i}; θ)

because its failure time exceeds

x_{i}

. Terms from the censoring distribution then factor out of the survival-parameter likelihood. Thus the survival likelihood and its logarithm are

L (θ) \propto \prod_{i = 1}^{n} f_{i} (x_{i}; θ)^{v_{i}} F_{i} (x_{i}; θ)^{1 - v_{i}},

ℓ (θ) = i = 1 \sum n {v_{i} lo g f_{i} (x_{i}; θ) + (1 - v_{i}) lo g F_{i} (x_{i}; θ)} + constant .

The individual subscripts allow the same parameter vector to act through different covariate values.

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