Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 207 / 4 / b

Table of contents
- Solution b

Solution

Let

Y (t)

be the at-risk process and

N (t)

the counting process for observed events. Over a short interval, the multiplicative-intensity model gives

E {d N (t) ∣ F_{t -}} = Y (t) h (t), d t = Y (t), d H (t),

where

h

is the hazard function and

H

the cumulative hazard function. Solving this relation for the infinitesimal hazard increment suggests

d H (t) = d N (t) / Y (t)

. Summing over distinct event times gives the Nelson–Aalen estimator

H (t) = \sum_{j : a_{j} \leq t} \frac{d _{j}}{r _{j}},

where

d_{j}

events occur among

r_{j}

individuals at risk. Here there are no ties, so

d_{j} = 1

.

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