Skip to content

IAI Actuarial Core Principles · Risk Modelling and Survival Analysis

Estimation Procedures for Lifetime Distributions in CS2

Estimation procedures for lifetime distributions let you estimate survival, hazard and mortality rates from real data where some lives are censored or truncated. You write the likelihood, then use Kaplan-Meier for S(t), Nelson-Aalen for H(t), the Cox model for covariates, and deaths ÷ exposed to risk for rates.

What this chapter covers

This chapter is about one question: how do you estimate survival quantities from data when you do not see every life until death? Real data has lives that leave the study alive, lives that enter late, and lives observed for different lengths of time. The chapter gives you the tools to handle this without bias.

You start with the data problem: censoring and truncation. You then build the likelihood for such data. Everything else follows from it. The Kaplan-Meier estimator of S(t) and the Nelson-Aalen estimator of H(t) are non-parametric. The Cox proportional hazards model is semi-parametric and adds covariates. The last topic, exposed to risk, gives the practical route to estimating mortality rates such as μx and qx.

This chapter connects to the rest of CS2 in several places. It builds on the survival models chapter, where you meet the hazard rate, the survival function and the cumulative hazard. It links to the Markov models for multiple states, where the same idea of exposure and events gives transition intensities. It also feeds Paper B, where you may be asked to fit these estimators in R and read the output.

Survival models carry a large share of the 2026 CS2 syllabus, and this chapter is the part that turns theory into numbers. Questions here are very calculable: you build a Kaplan-Meier table, compute a Nelson-Aalen value, or write a likelihood from a short data set, and method marks are easy to earn if your layout is clean. The same ideas also appear in Paper B through R output, so the effort pays twice. Students who learn the logic of risk sets and exposure, rather than memorising formulas, usually find the whole chapter quick to revise.

Estimation procedures for lifetime distributions: topics in the order to study them

  1. 1Censoring and Truncation in Lifetime DataEvery later estimator exists to deal with incomplete observation, so you must know the types (right, left, interval censoring; left and right truncation) first.
  2. 2Likelihood Functions for Censored Survival DataThe likelihood shows how a death and a censored life each contribute, and this logic sits behind all the estimators that follow.
  3. 3Kaplan-Meier Estimator of the Survival FunctionIt is the most tested estimator and uses the risk set idea that you need for Nelson-Aalen and Cox.
  4. 4Nelson-Aalen Estimator of the Cumulative HazardIt uses the same table as Kaplan-Meier, so you learn it quickly and can compare the two estimates.
  5. 5Cox Proportional Hazards Model EstimationIt needs the hazard, risk set and partial likelihood ideas from the earlier topics, so it comes after them.
  6. 6Estimating Mortality Rates from Exposed to RiskIt applies the same events-over-exposure idea to grouped data, and it links to the Poisson and binomial models you may already know.

How to prepare Estimation procedures for lifetime distributions

Work from the data problem to the estimators, and practise each one on a small table by hand before you touch R.

  1. Define each type of censoring and truncation in your own words. For each, say what you know about the life and what you do not.
  2. Write the likelihood for a small data set: a density term f(t) for each death and a survival term S(t) for each right-censored life. Do this until it is automatic.
  3. Build Kaplan-Meier tables with columns for time, number at risk, deaths, censored and the running product of (1 − d ÷ n). Check that the risk set falls correctly after each time.
  4. Add a Nelson-Aalen column to the same table, summing d ÷ n. Compare exp(−H) with Kaplan-Meier to see how close they are.
  5. Learn the Cox model structure: h(t; z) = h0(t) exp(β·z). Know what the partial likelihood uses, how to read a hazard ratio exp(β), and what the proportional hazards assumption means.
  6. Practise exposed-to-risk questions with both a census approach and a rate-based approach, and state your assumption about when deaths occur within the year.
  7. Run the R functions for this chapter on a small data set. Be ready to explain the output in words, not just copy it.

Common mistakes in Estimation procedures for lifetime distributions

  • Treating censored lives as deaths, or dropping them from the data altogether.

    Fix: Keep censored lives in the risk set up to their censoring time. They add to the denominator but never to the deaths.

  • Getting the number at risk wrong in a Kaplan-Meier or Nelson-Aalen table.

    Fix: Update the risk set row by row, and check that nj equals nj−1 − dj−1 − cj−1. Treat a censoring at tj as still at risk at tj, as is standard, and state this.

  • Confusing the likelihood contribution of truncation with that of censoring.

    Fix: Remember the difference: censoring changes what you see about the end of a life, while truncation changes which lives you get to see at all. Truncation means conditioning on survival to entry.

  • Interpreting the Cox coefficient β as the hazard ratio.

    Fix: Always compute exp(β). A positive β gives a ratio above 1, so a higher hazard. State the covariate change the ratio refers to.

  • Mixing up central and initial exposed to risk, or using the wrong one for the rate.

    Fix: Link them to the target: central exposure goes with μ, initial exposure goes with q. If the formula does not match, convert using a stated assumption.

  • Giving a numerical answer for a Paper B question with no explanation of the method or assumptions.

    Fix: Say which estimator you used, what the output shows, and what it means for the survival or hazard in context.

Last-day revision: Estimation procedures for lifetime distributions

  • Right censoring: the life is known to survive past a time, but the exact lifetime is unknown. Left truncation: lives enter observation only after surviving to some age.
  • Likelihood: deaths contribute f(t), right-censored lives contribute S(t), and left-truncated lives are conditioned on survival to entry.
  • Kaplan-Meier: Ŝ(t) = Π over death times tj ≤ t of (1 − dj ÷ nj).
  • nj is the number at risk just before tj, so it includes lives censored at tj but excludes those who left earlier.
  • Nelson-Aalen: Ĥ(t) = Σ over tj ≤ t of dj ÷ nj, and Ŝ(t) = exp(−Ĥ(t)) is the matching survival estimate.
  • Kaplan-Meier changes only at death times and is a step function that is constant between them.
  • Cox model: h(t; z) = h0(t) exp(β·z), where the baseline hazard h0(t) is left unspecified.
  • The hazard ratio for two lives is exp(β·(z1 − z2)) and does not depend on time under proportional hazards.
  • The Cox partial likelihood at each death compares the failing life's hazard with the total over the risk set.
  • Central exposed to risk Ec is the total time lived in the age group, and the estimate of μ is deaths ÷ Ec.
  • Initial exposed to risk E is used for qx, and the estimate of q is deaths ÷ E.
  • Always state the assumption you use, such as constant force of mortality over the age interval.

Estimation procedures for lifetime distributions practice questions

Estimation procedures for lifetime distributions: frequently asked questions

Do I need to memorise the Kaplan-Meier and Nelson-Aalen formulas?

Yes, both are short and used often, so learn them. Ŝ(t) = Π(1 − dj ÷ nj) and Ĥ(t) = Σ dj ÷ nj. It helps more to understand why the risk set sits in the denominator than to memorise the shape.

How do Kaplan-Meier and Nelson-Aalen differ?

Kaplan-Meier estimates the survival function directly as a product. Nelson-Aalen estimates the cumulative hazard as a sum. You can turn the Nelson-Aalen estimate into a survival estimate with exp(−Ĥ(t)), and the two are usually close.

How much of the Cox model do I need for CS2?

Know the model form, the meaning of the baseline hazard, how to read and use hazard ratios, and what the partial likelihood is built from. Be able to read R output too. Fitting by hand is usually limited to simple cases.

Is this chapter tested in Paper B as well?

Survival analysis is a natural fit for the computer-based paper, so be ready to fit and interpret these estimators in R. Check the latest IAI syllabus and past papers to confirm the scope.