Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 207 / 5 / b / vii

Insert the Breslow estimator into the fitted cumulative intensities and interchange the finite sums:

i = 1 \sum n e^{β z_{i}} H_{0} (x_{i}) = i = 1 \sum n e^{β z_{i}} ℓ : x_{ℓ} \leq x_{i} \sum \frac{v _{ℓ}}{\sum _{j : x_{j} \geq x_{ℓ}} e ^{β z_{j}}} = ℓ = 1 \sum n \frac{v _{ℓ} \sum _{i : x_{i} \geq x_{ℓ}} e ^{β z_{i}}}{\sum _{j : x_{j} \geq x_{ℓ}} e ^{β z_{j}}} = ℓ = 1 \sum n v_{ℓ} .

(1)

Every observed event contributes exactly one after summing its fitted contribution over its risk set. Therefore

i = 1 \sum n y_{i} = i \sum v_{i} - i \sum e^{β z_{i}} H_{0} (x_{i}) = 0.

(2)

This identity is algebraic and holds for any finite coefficient used consistently in the relative hazards and the Breslow estimator; it does not require the score equation to be solved exactly. It can fail if the baseline hazard is estimated or smoothed by a different procedure.

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Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 207 / 5 / b / vii

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