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

Block maxima of daily losses are modelled by a GEV distribution with shape ξ > 0 after suitable normalisation. Which statement is correct?

A positive shape parameter gives the Fréchet type, with a heavy polynomial tail decaying like x to the power minus 1/ξ. Consequently moments of order 1/ξ and above are infinite. Gumbel has ξ equal to zero and Weibull has negative ξ.

  1. AThe distribution has a finite upper endpoint and belongs to the Weibull domain
  2. BThe distribution is the Gumbel type with exponentially decaying tail
  3. CThe distribution has a heavy, polynomially decaying tail of Fréchet type, and moments of order 1/ξ or higher do not existCorrect
  4. DThe distribution is the Fréchet type and all moments exist
  5. The shape parameter ξ equals the number of observations in each block

Explanation

ξ > 0 corresponds to the Fréchet case, with tail behaving like x^(-1/ξ). Hence moments of order at or above 1/ξ are infinite. ξ = 0 is Gumbel and ξ < 0 is Weibull with a finite endpoint.

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