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

A GEV distribution has shape parameter ξ = 0.4. Which statement about the tail of this distribution is correct?

With ξ = 0.4 > 0 the GEV is of Fréchet type, with a heavy polynomial tail. Moments of order 1/ξ = 2.5 or above are infinite, so, for example, the third moment does not exist. Weibull needs a negative shape and Gumbel needs zero.

  1. AIt is of Weibull type with a finite upper endpoint
  2. BIt is of Gumbel type with an exponentially decaying tail
  3. CIt is of Fréchet type with a heavy tail, so some higher moments are infiniteCorrect
  4. DIt is of Fréchet type, and all moments are finite
  5. It is of Weibull type with a heavy tail

Explanation

A positive shape parameter gives the Fréchet type, which has a polynomially decaying (heavy) tail. Moments of order 1/ξ or higher do not exist; here 1/0.4 = 2.5, so the third moment is infinite. Weibull requires ξ < 0 and Gumbel is ξ = 0.

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