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

For the constant-force model, the MLE mu-hat = D/V where D is the number of deaths and V the total waiting time. Which statement about its properties is correct for large samples?

The MLE is asymptotically normal, centred on the true mu, with variance about mu squared divided by the expected number of deaths. It is not exactly unbiased in finite samples, and its variance decreases as expected deaths increase.

  1. AIt is asymptotically normal with mean mu and variance approximately mu^2/E[D]Correct
  2. BIt is exactly unbiased in every finite sample
  3. CIt has a chi-squared distribution with D degrees of freedom
  4. DIt is asymptotically normal with mean mu and variance approximately mu/E[D]
  5. It has variance independent of the number of deaths

Explanation

The Cramer-Rao asymptotic variance is mu^2/E[D], since the information is E[D]/mu^2. Hence the variance falls as expected deaths rise. The estimator is biased in finite samples, and a variance of mu/E[D] has the wrong dimensions.

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