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IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Estimation procedures for lifetime distributions

An insurer studies mortality using only policies that were in force on 1 January 2020, with ages recorded at that date. Lives who died before that date are absent from the data. What is this feature called, and what is its correct treatment?

This is left truncation. Lives who died before the study began are never observed, so each life's likelihood contribution must be conditioned on survival to its entry age, dividing by the survival function at entry.

  1. ARight censoring; include the lives with a survival probability of one
  2. BLeft truncation; condition each life's likelihood contribution on survival to its age at entryCorrect
  3. CLeft censoring; treat entry age as the age at death
  4. DInformative censoring; discard all lives entering after age 30
  5. Right truncation; use only the lives that died in the study

Explanation

Lives dying before the study start never enter the sample, so the data is left truncated. Each life's likelihood is conditional on having survived to entry age, giving f(x)/S(entry) for deaths and S(x)/S(entry) for censored lives. Ignoring this biases survival estimates upward.

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