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

For the constant-force model with D deaths and total waiting time V, the Cramér–Rao lower bound for an unbiased estimator of μ is approximately μ/E[V]. What is the key reason this arises from the likelihood?

The log-likelihood is D ln μ − μV, whose second derivative is −D/μ². Because E[D] = μE[V], the Fisher information is E[V]/μ, and its inverse gives the variance bound μ/E[V].

  1. AThe second derivative of the log-likelihood is −D/μ², and its expectation is −E[V]/μCorrect
  2. BThe log-likelihood is linear in μ
  3. CThe likelihood is maximised at μ = V/D
  4. DThe first derivative of the log-likelihood is always positive
  5. The score has expectation equal to E[V]

Explanation

The log-likelihood is D ln μ − μV. The second derivative is −D/μ². Since E[D] = μE[V], the expected value is −E[V]/μ. The inverse of the information gives variance μ/E[V].

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