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

Which statement about the MLE μ-hat = D/V for a constant force of mortality is correct?

The MLE D/V is asymptotically normal and asymptotically unbiased, though it can be biased in small samples. Its variance depends on the true μ, and it is consistent, so it is not always below or above μ.

  1. AIt is exactly unbiased for every sample size
  2. BIt is asymptotically normal and asymptotically unbiased, but may be biased in small samplesCorrect
  3. CIt has a variance that does not depend on the true μ
  4. DIt is always smaller than the true μ
  5. It is inconsistent because V is random

Explanation

Standard MLE theory gives asymptotic normality, asymptotic unbiasedness and consistency as the amount of data grows. In finite samples the ratio D/V need not be exactly unbiased. Its asymptotic variance μ²/E[D] clearly depends on μ.

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