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

In a continuous-time multiple-state model, the likelihood for a single life observed over a period is built from the time spent in each state and the transitions made. Which statement about the maximum likelihood estimator of a constant transition intensity from state i to state j is correct?

The maximum likelihood estimator of a constant transition intensity from i to j is the observed number of transitions from i to j divided by the total waiting time all lives spent in state i, because exposure time is the relevant denominator.

  1. AIt equals the number of transitions from i to j divided by the total time spent in state i by all livesCorrect
  2. BIt equals the number of transitions from i to j divided by the number of lives that ever entered state i
  3. CIt equals the total time spent in state i divided by the number of transitions from i to j
  4. DIt equals the number of transitions from i to j divided by the total time spent in state j
  5. It equals the proportion of all exits from state i that went to state j, ignoring time

Explanation

With constant intensities the likelihood is a product of terms exp(-sum of mu_ik times waiting time) times mu_ij to the power of transition count. Maximising gives mu_ij hat = d_ij / V_i, where V_i is total time in state i. Dividing by the number of lives ignores exposure time, and the time in state j is irrelevant.

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