IAI Actuarial Core Principles · Risk Modelling and Survival Analysis

Maximum Likelihood Estimators for Transition Intensities

A transition intensity μ is the rate of moving between states. With a constant intensity, the MLE is μ̂ = number of transitions observed ÷ total time spent in the starting state (central exposure). Its variance is approximately μ² ÷ number of transitions. In multi-state models, you estimate each intensity separately the same way.

What this chapter covers

This chapter in CS2 shows how to estimate transition intensities from observed data. You start with the simplest case, the two-state alive-dead model. You assume the force of mortality μ is constant over the period of study. You then write down the likelihood from the data. Each life contributes the time it was observed, and each death contributes a factor of μ.

The key result is simple. The MLE of μ is the number of deaths divided by the total waiting time, which is the total time lives were exposed to risk. The chapter then shows how to get the variance of this estimator from the second derivative of the log-likelihood, or from the Cramér-Rao lower bound. You use this to build confidence intervals. After that you extend the idea to models with several states, such as healthy, sick and dead, where each intensity has its own count and its own waiting time.

This chapter connects to the rest of CS2 in several ways. It builds on the Markov jump process material, where transition intensities define the model. It sits beside the survival models topics, such as the Kaplan-Meier and Cox models, which estimate hazards without assuming a constant rate. It also uses maximum likelihood ideas from CS1. In Paper B, you may need to compute these estimators in R from a data set.

Survival models carry a large share of the CS2 syllabus, and this chapter is one of its most formula-driven and predictable parts. Questions are often short and numerical, so you can score full marks if you set up the likelihood correctly and show your working. The same ideas also support questions on multiple-state models, confidence intervals and the link between observed data and model parameters. Written questions often ask you to derive the estimator, so understanding why it works earns more than memorising it. IAI sets the pass mark rules for each session, so check them when results are published.

Maximum likelihood estimators for transition intensities: topics in the order to study them

1Two-State Model and Observed Data SetupYou need to know what data you observe, such as deaths and waiting time, and what assumptions you make before any likelihood makes sense.
2Maximum Likelihood Estimator of Transition IntensityOnce the data setup is clear, you can build the likelihood and derive μ̂ = deaths ÷ total waiting time.
3Properties and Variance of the MLEThis topic needs the estimator first, since you find its variance and distribution from the log-likelihood and then use them for confidence intervals.
4Multiple-State Models and Estimating Several IntensitiesThis topic extends the same method to each transition, so it works best once the two-state case is secure.

How to prepare Maximum likelihood estimators for transition intensities

Treat this chapter as one derivation that you learn once and then reuse. Aim to be able to write it from a blank page.

Write out the two-state setup in your own words: the states, the constant intensity assumption, the observed deaths and the total time observed for each life.
Derive the likelihood by hand. Each life contributes exp(−μ × time observed), and each death adds a factor of μ. Take logs, differentiate, set to zero and solve for μ̂.
Check the second derivative to show it is a maximum. Then derive the variance and note the asymptotic result: μ̂ is approximately normal with mean μ and variance μ² ÷ number of deaths.
Practise building a 95% confidence interval for μ with the estimated variance. Do several numerical questions with different data until the steps are automatic.
Extend to a three-state model. For each transition, find the count of that transition and the total time spent in the starting state, then divide. Practise explaining why the likelihood splits into separate parts.
Do past-paper questions under time limits. Then repeat the main calculation in R so you are ready for Paper B.

Common mistakes in Maximum likelihood estimators for transition intensities

Dividing the number of deaths by the number of lives instead of by the total time observed.
Fix: Always compute the total waiting time first, adding the time each life was observed. Then divide the count by that total.
Using the wrong exposure in a multi-state model.
Fix: For each intensity μij, use only the time spent in state i. Write that total next to each transition before dividing.
Skipping the derivation and the check that the estimate is a maximum.
Fix: Show the log-likelihood, the first derivative set to zero, and the sign of the second derivative. Written questions award marks for these steps.
Quoting the variance without stating that it is an asymptotic approximation.
Fix: Say that it holds approximately for large samples. Use the estimated version, μ̂² ÷ d, when building numerical confidence intervals.
Forgetting to state the assumptions, such as constant intensity over the period.
Fix: Open each answer with one line on the model and the assumptions. This also helps you judge whether the model suits the data.

Last-day revision: Maximum likelihood estimators for transition intensities

Two-state model: alive to dead with constant intensity μ over the period of study.
Data needed: number of deaths and the total time each life was observed.
The total observed time is called the waiting time or central exposure.
Likelihood: L(μ) ∝ μ^d × exp(−μ × v), where d = deaths and v = total waiting time.
MLE: μ̂ = d ÷ v.
Variance: approximately μ² ÷ E[D], estimated as μ̂² ÷ d.
Large samples: μ̂ is approximately normal with mean μ and variance μ² ÷ E[D].
Approximate 95% confidence interval: μ̂ ± 1.96 × μ̂ ÷ √d.
Multi-state: for each transition i to j, μ̂ij = number of i to j transitions ÷ total time in state i.
In a multi-state model, the likelihood factorises, so each intensity is estimated separately.
State your assumptions: constant intensity, and independent lives.

Maximum likelihood estimators for transition intensities practice questions

Maximum likelihood estimators for transition intensities: frequently asked questions

What is the MLE of a transition intensity in a two-state model?

If the intensity μ is constant, the MLE is μ̂ = d ÷ v. Here d is the number of observed transitions, such as deaths, and v is the total time lives were observed in the starting state. You derive it by maximising the likelihood.

How do I find the variance of the MLE?

Use the second derivative of the log-likelihood, or the Cramér-Rao lower bound. For large samples, the variance of μ̂ is approximately μ² ÷ E[D]. In practice you replace it with μ̂² ÷ d.

How do I estimate several intensities in a multi-state model?

Estimate each transition separately. For a transition from state i to state j, divide the number of such transitions by the total time spent in state i. The likelihood splits into separate parts, which makes this valid.

Do I need R for this chapter?

Written Paper A questions test the derivation and calculation by hand. In Paper B, you may be asked to compute the estimates from data in R. Practise both so you can explain the method and run it.