IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Maximum likelihood estimators for transition intensities
In a two-state model (Alive to Dead) with constant force of mortality mu, a study observes n lives. The total observed time exposed to risk is E years and d deaths occur. Which is the maximum likelihood estimator of mu?
The MLE of a constant force of mortality is the number of deaths divided by the total time exposed to risk, d/E. This follows from maximising d ln(mu) minus mu E. The reciprocal E/d would estimate mean waiting time, not the intensity.
- Ad/ECorrect
- BE/d
- Cd/(n*E)
- Dd*E
- n/E
Explanation
The likelihood is proportional to mu^d * exp(-mu*E). Taking logs gives d*ln(mu) - mu*E, and setting the derivative to zero gives mu-hat = d/E. E/d is the reciprocal, which estimates the mean lifetime rather than the intensity.
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