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IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Maximum likelihood estimators for transition intensities

In a three-state model (Healthy H, Sick S, Dead D) with constant transition intensities, the observed data over a period show 40 transitions from H to S, with total time spent in state H equal to 2,000 years. What is the MLE of the intensity from H to S, and what feature of the likelihood does it rely on?

The MLE is 40 divided by 2,000, which is 0.02. The likelihood factorises over transition types, and the exposure for each intensity is the total time spent in the originating state, whatever the destination.

  1. A0.020, using time spent in H as exposure regardless of the destination stateCorrect
  2. B0.020, using only time of those who moved from H to S
  3. C0.025, using time spent in S as exposure
  4. D0.050, using time spent in H divided by transitions
  5. 0.020, using total time spent in all three states

Explanation

The estimate is 40/2000 = 0.02. The likelihood factorises by transition type, and each intensity's exposure is the total time spent in the originating state, whatever the eventual destination. Using only movers' time or all states' time is incorrect.

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