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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 μ, the data give the number of deaths d and the total time spent exposed to risk v. Which statement about the maximum likelihood estimator μ̂ = D/V is correct?

The MLE μ̂ = D/V is asymptotically normally distributed with mean μ. It is not exactly unbiased in finite samples, and its variance depends on μ, so the asymptotic normality statement is the only correct one among the options.

  1. AIt is exactly unbiased for μ for every sample size
  2. BIt is asymptotically normally distributed with mean μCorrect
  3. CIt is always equal to the proportion of lives that die
  4. DIt has variance that does not depend on μ
  5. It is a biased estimator with no large-sample distribution

Explanation

The MLE of a constant force is deaths divided by central exposed to risk. It is not exactly unbiased in general but is asymptotically normal with mean μ and variance μ/E[V]. The proportion dying is a different quantity, so it is not the estimator.

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