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IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Estimation procedures for lifetime distributions

Ten lives are observed from birth under an exponential model with force of mortality μ. Four die, at times 2, 3, 5 and 6 years. The other six are withdrawn alive at the end of year 8 (Type I censoring). What is the maximum likelihood estimate of μ?

The estimate is 0.0625. The exponential MLE equals the number of deaths divided by total observed time. Total time is 16 for the deaths plus 48 for the six censored lives, giving 64 years, so 4/64 equals 0.0625.

  1. A4/64 = 0.0625Correct
  2. B4/16 = 0.2500
  3. C4/48 = 0.0833
  4. D4/10 = 0.4000
  5. 6/64 = 0.0938

Explanation

The MLE is deaths divided by total time exposed. Total exposure = 2+3+5+6 + 6×8 = 16 + 48 = 64 years. Deaths = 4, so μ-hat = 4/64 = 0.0625. Using only the 48 years of censored lives, or only deaths' times, ignores part of the exposure.

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