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IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Estimation procedures for lifetime distributions

Which statement correctly describes how the Nelson-Aalen estimate of the cumulative hazard, Λ̂(t), is used to obtain a survival function estimate?

The survival estimate is exp(−Λ̂(t)). This follows from the relationship S(t)=exp(−Λ(t)). It is generally not identical to the Kaplan-Meier estimate, which multiplies (1 − d/n) terms, although the two are close when risk sets are large.

  1. AŜ(t) = exp(−Λ̂(t)), and this generally differs from the Kaplan-Meier estimateCorrect
  2. BŜ(t) = 1 − Λ̂(t), and it equals the Kaplan-Meier estimate
  3. CŜ(t) = exp(Λ̂(t)), which is always greater than 1
  4. DŜ(t) = Λ̂(t)/t, which equals the Kaplan-Meier estimate
  5. Ŝ(t) = 1/(1+Λ̂(t)), which equals the Kaplan-Meier estimate

Explanation

Since S(t)=exp(-Λ(t)), the Nelson-Aalen based estimate is exp(-Λ̂(t)). The Kaplan-Meier product of (1-d/n) differs, because exp(-d/n) is not equal to 1-d/n, though they are close when d/n is small.

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