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

In a two-state alive-dead model with a constant force of mortality μ, the data show d deaths and a total time spent alive (central exposed to risk) of v. Which statement about the maximum likelihood estimator of μ is correct?

The MLE of a constant force of mortality is the number of deaths divided by the total time lived (central exposed to risk), d/v. This follows from maximising the likelihood μ^d e^(−μv). It does not require equal observation periods for each life.

  1. AIt is d/v, and it is the ratio of observed deaths to total waiting timeCorrect
  2. BIt is v/d, and it is the average time lived per death
  3. CIt is d/(v+d), because deaths occur on average halfway through the period
  4. DIt is d/v, but only if every life is observed for exactly one year
  5. It is 1 − d/v, the complement of the observed death rate

Explanation

The likelihood is proportional to μ^d e^(−μv). Taking logs gives d ln μ − μv, and setting the derivative to zero gives μ-hat = d/v. The estimator needs no fixed observation period, so the option requiring one year of observation is wrong.

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