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IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Transition intensities dependent on age (exact or census)

Under the constant force of mortality model, the MLE mu-hat = D/V is used. For large exposure, which statement describes the asymptotic distribution of mu-hat?

For large exposure the estimator is approximately normal with mean mu and variance mu divided by the expected total exposure E[V]. This follows from the Cramér-Rao lower bound, using the expected information E[V]/mu from the log-likelihood.

  1. AApproximately normal with mean mu and variance mu/E[V]Correct
  2. BApproximately normal with mean mu and variance mu squared times E[V]
  3. CExactly exponential with mean mu
  4. DApproximately normal with mean 1/mu and variance mu/E[V]
  5. Approximately Poisson with mean mu

Explanation

The second derivative of the log-likelihood is -d/mu^2, with expectation -E[V]/mu. The Cramér-Rao bound gives variance mu/E[V]. The MLE is asymptotically unbiased and normal, so the other options misstate the mean or variance.

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