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IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Loss distributions, with and without risk sharing

Claims below a policy limit are fully observed, but claims at the limit are only known to equal or exceed it. For an exponential model fitted by maximum likelihood, what is the contribution of a claim censored at limit L to the log-likelihood?

A censored claim contributes the survival probability P(X ≥ L) = e^(−λL) to the likelihood. Its log-likelihood contribution is therefore −λL. Including ln λ would treat the claim as an exactly observed value, which it is not.

  1. Aln λ − λL
  2. B−λLCorrect
  3. CλL
  4. Dln(λ) + λL
  5. −ln(λL)

Explanation

A censored observation contributes P(X ≥ L) = e^(−λL), the survival function. Its log is −λL. The term ln λ − λL would be the contribution of an exactly observed claim at L, which wrongly treats the censored claim as an exact value.

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