 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2010/iii/paper-34/3/a/solution

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

Assume independent individuals and independent censoring, with the censoring law not involving the event-rate parameters. If

T_{j}

C_{j}

its censoring time, we observe

x_{j} = min (T_{j}, C_{j})

and

v_{j} = 1_{{T_{j} \leq C_{j}}}

f_{j} (x) = θ_{j} e^{- θ_{j} x}, S_{j} (x) = e^{- θ_{j} x} .

An observed event contributes

f_{j} (x_{j})

S_{j} (x_{j})

, because it only says the event has not occurred by

x_{j}

. Multiplying these survival likelihood contributions and omitting censoring factors independent of the parameters gives

L (θ_{1}, \dots, θ_{n}) \propto \prod_{j = 1}^{n} θ_{j}^{v_{j}} e^{- θ_{j} x_{j}}, ℓ (θ) = j = 1 \sum n {v_{j} lo g θ_{j} - θ_{j} x_{j}} + constant .

Without independent censoring the censoring factors cannot generally be discarded in this way.

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