Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 29 / 2 / b / ii

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- Solution ii

Solution

Insert

f_{i} = h_{i} F_{i}

and

F_{i} = e^{- H_{i}}

into the survival likelihood. Each individual's log contribution simplifies to

v_{i} lo g h_{i} (x_{i}; θ) + lo g F_{i} (x_{i}; θ) .

Therefore

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

Failures contribute a log hazard function term as well as exposure, whereas both failures and censored subjects contribute the negative cumulative hazard function over their observed follow-up.

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