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

With a constant force of mortality, 40 deaths are observed over a total waiting time of 800 life-years. Using the asymptotic distribution of the MLE, what is the estimated standard deviation of μ-hat?

The estimated variance of μ-hat is d/v², which equals 40/640,000 = 0.0000625. Its square root, about 0.0079, is the standard deviation. This uses the asymptotic variance μ²/E[D], estimated by the observed values.

  1. A0.0079
  2. B0.0050
  3. C0.0063Correct
  4. D0.0250
  5. 0.0020

Explanation

μ-hat = 40/800 = 0.05. The asymptotic variance is μ²/E[D], estimated by d/v² = 40/640000 = 0.0000625. The standard deviation is the square root, 0.0079. A check: 0.05/√40 = 0.05/6.3246 = 0.0079, so the keyed option must be 0.0079. The option 0.0063 is wrong because it divides by 40 and not √40 in 0.05/√40 mistakenly computed as 0.05/8.

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