IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Maximum likelihood estimators for transition intensities
In the two-state alive-dead model with constant force of mortality μ, observations are made on lives over a period. Let D be the number of deaths and V the total time spent alive (waiting time) by all lives. What is the maximum likelihood estimator of μ?
The MLE of a constant force of mortality in the two-state model is the number of deaths divided by the total time lived by all lives. This follows from the likelihood proportional to μ^D exp(-μV), maximised at D/V.
- AD divided by the number of lives at the start
- BV divided by D
- CD divided by VCorrect
- DD multiplied by V
- D divided by (V plus D)
Explanation
The likelihood is proportional to exp(-μV) μ^D. Taking logs gives D ln μ - μV, and setting the derivative to zero gives μ = D/V. V/D is the reciprocal, which estimates mean lifetime, not the force.
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