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

In the two-state alive-dead model with a constant force of mortality μ, a study observes n lives. Let D be the number of deaths and V the total time spent alive (waiting time) under observation. Which expression is the maximum likelihood estimator of μ?

The MLE is the number of deaths divided by the total waiting time lived, D/V. The likelihood is proportional to μ^D e^(-μV), and maximising its logarithm gives μ̂ = D/V. The reciprocal V/D estimates the mean survival time instead.

  1. AD / n
  2. BV / D
  3. CD / VCorrect
  4. Dn / V
  5. D / (n × V)

Explanation

The likelihood for the constant-force model is proportional to μ^D · exp(-μV). Taking logs gives D ln μ - μV, and setting the derivative D/μ - V to zero yields μ̂ = D/V. V/D is the reciprocal, which estimates mean lifetime rather than the force.

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