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IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Introduction to extreme value theory

In a peaks-over-threshold analysis, the fitted GPD has an estimated shape parameter ξ = −0.2 with a 95% confidence interval entirely below zero. What is the most appropriate conclusion about the tail of the loss distribution?

A significantly negative shape parameter means the excesses come from a short-tailed GPD with a finite upper bound at u − β/ξ. Heavy tails and infinite moments occur only for positive shape, and the exponential case needs ξ equal to zero.

  1. AThe tail is heavy like a Pareto with infinite variance
  2. BThe tail is exponential, exactly as in the Gumbel domain
  3. CThe distribution of excesses has a finite upper endpoint, at u − β/ξCorrect
  4. DThe mean excess function is increasing linearly in the threshold
  5. The fitted model has no finite mean

Explanation

A negative shape parameter means the GPD has bounded support, with upper endpoint u − β/ξ = u + 5β. Mean excess then decreases linearly with the threshold, and all moments exist.

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