At time $x_i$, the observed number of events is $v_i$ and the exposure to the common instantaneous hazard is the risk-set size $n-i+1$. The likelihood score for a hazard increment therefore equates observed and expected events:
$$
v_i=(n-i+1)\,d\widehat H(x_i).
$$
Thus the <Nelson–Aalen estimator> is
$$
\widehat H(t)
=\sum_{i:x_i\leq t}\frac{v_i}{n-i+1}.
$$
It estimates the <cumulative hazard function> by adding event count divided by current exposure at every observed event time.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 207 / 4 / a / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-207/4/a/ii/solution