Skip to content

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

A two-state model with constant μ is fitted to data with D deaths. The observed data for each life i consist of the time θ_i observed and an indicator d_i of death (1) or censoring (0). Which statement about the likelihood and the estimator is correct?

A censored life contributes only the survival probability exp(-μθ_i) to the likelihood. Its observed time is added to the total waiting time V, but it does not count as a death in D. Ignoring censored lives would overstate the force of mortality.

  1. ACensored lives contribute exp(-μθ_i) to the likelihood and their time is included in V, but they do not add to DCorrect
  2. BCensored lives are excluded from both D and V
  3. CCensored lives contribute μ·exp(-μθ_i) to the likelihood
  4. DCensored lives add to D but not to V
  5. The MLE equals the mean of θ_i over all lives

Explanation

The likelihood is the product of μ^d_i · exp(-μθ_i). A censored life has d_i=0, so contributes only the survival factor exp(-μθ_i). Its time enters V but it adds nothing to D. Excluding censored lives would bias μ̂ upwards.

Did you get it right without looking?

One question tells you little. A timed set on Maximum likelihood estimators for transition intensities shows your real accuracy, how long you take and where you lose marks.

Practise Maximum likelihood estimators for transition intensitiesFree diagnostic mock

More Maximum likelihood estimators for transition intensities questions