In the Nelson-Aalen estimator, the cumulative hazard H(t) is estimated as the sum at each event time of which term?

• A. Sum of d_j divided by r_j
• B. Sum of r_j divided by d_j
• C. Product of (1 - d_j / r_j)
• D. Difference of d_j and r_j

Answer: (A)

The main idea is that the Nelson-Aalen estimator builds the cumulative hazard by adding a small hazard contribution at each observed event time. At an event time t_j, that contribution is the number of events observed there, d_j, divided by the number at risk just before that time, r_j. So each event time adds d_j / r_j to the cumulative hazard. Summing these increments up to time t gives the estimator: H_hat(t) = sum over event times t_j ≤ t of d_j / r_j.

This is why the correct form is the sum of d_j divided by r_j. The other options don’t match how the cumulative hazard is constructed: using r_j/d_j would invert the increment, the product of (1 - d_j / r_j) is the Kaplan-Meier estimator for survival (not cumulative hazard), and a simple difference d_j − r_j has no meaningful interpretation as a hazard contribution.