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

In a multiple-state model with a Markov assumption and constant transition intensities, the maximum likelihood estimator of the intensity from state i to state j is based on which pair of observed quantities?

The estimator is the observed number of transitions from state i to state j divided by the total time all lives spent in state i. This arises from maximising the likelihood with exponential waiting times, so exposure time rather than head count is the correct denominator.

  1. ANumber of transitions from i to j divided by the total time spent in state iCorrect
  2. BNumber of transitions from i to j divided by the number of lives ever in state i
  3. CTotal time spent in state i divided by the number of transitions from i to j
  4. DNumber of transitions from i to j divided by the total time spent in all states
  5. Number of transitions out of state i to any state divided by the number of lives entering i

Explanation

With constant intensities the likelihood contains a factor exp(-mu_ij v_i) mu_ij^(d_ij), where v_i is total waiting time in state i. Maximising gives mu_ij = d_ij / v_i. Dividing by the number of lives ignores exposure time and is wrong.

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