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

A study of 400 policyholders records 20 deaths with a total of 2,500 life-years of central exposure. Assuming a constant force of mortality, what are the MLE of mu and its estimated standard deviation (using the asymptotic variance mu/E[V])?

The estimate is 20/2500 = 0.008. The variance is approximately d/v squared, 20/6,250,000 = 3.2e-6, giving a standard deviation of about 0.00179. The 400 lives are headcount, not time exposed, so they are not the denominator.

  1. Amu-hat 0.0080, standard deviation about 0.00179Correct
  2. Bmu-hat 0.0500, standard deviation about 0.0112
  3. Cmu-hat 0.0080, standard deviation about 0.0032
  4. Dmu-hat 0.0500, standard deviation about 0.0045
  5. mu-hat 0.0080, standard deviation about 0.0089

Explanation

mu-hat = 20/2500 = 0.008. Estimated variance = mu-hat/v = 0.008/2500 = 3.2e-6, equivalently d/v^2. The square root is 0.00179. Using 400 as exposure gives 0.05, which is wrong.

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