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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 constant force of mortality mu, a study observes n lives, with total observed time exposed to risk v and d deaths. Which expression is the maximum likelihood estimator of mu?

The MLE is the number of deaths divided by the total time lived by all individuals, d/v. The likelihood is proportional to mu^d exp(-mu v), and maximising it gives this central exposure rate for a constant force of mortality.

  1. Ad / vCorrect
  2. Bv / d
  3. Cd / n
  4. Dn / (d + v)
  5. d / (n v)

Explanation

The likelihood is proportional to mu^d * exp(-mu v). Taking logs gives d ln(mu) - mu v, and setting the derivative d/mu - v to zero gives mu-hat = d/v. The option v/d is the reciprocal, which estimates mean waiting time, not the intensity.

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