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

In a two-state alive-dead model with a constant force of mortality mu, the data record the number of deaths d and the total time exposed to risk v (central exposed to risk). What is the maximum likelihood estimator of mu?

The maximum likelihood estimator is deaths divided by total time exposed to risk, d/v. The likelihood is proportional to mu^d times exp(-mu v), and maximising its logarithm gives d/v, the observed occurrence rate per unit of exposure.

  1. Ad divided by vCorrect
  2. Bv divided by d
  3. Cd divided by (d + v)
  4. Dd multiplied by v
  5. d divided by v squared

Explanation

The likelihood is proportional to mu^d exp(-mu v). Taking logs gives d ln mu - mu v. Differentiating and setting to zero gives mu-hat = d/v. The option v/d inverts the ratio and estimates the mean lifetime rather than the intensity.

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