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

Block maxima of a loss variable are fitted with a generalised extreme value distribution with shape parameter ξ. For which value of ξ is the underlying loss distribution heavy tailed, in the Fréchet domain of attraction with no finite upper bound and polynomial tail decay?

A positive shape parameter ξ indicates the Fréchet domain, where the loss distribution has a heavy, power-law tail and no upper bound. ξ = 0 gives Gumbel with exponential-type tails, and negative ξ gives Weibull with a finite upper endpoint.

  1. Aξ = 0
  2. Bξ < 0
  3. Cξ > 0Correct
  4. Dξ = -1
  5. ξ is irrelevant to the tail

Explanation

A positive shape parameter corresponds to the Fréchet case, with tails decaying like a power function and infinite upper endpoint. ξ = 0 is Gumbel (exponential-type tails) and ξ < 0 is Weibull with a finite upper endpoint.

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