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

The Generalised Extreme Value (GEV) distribution has distribution function G(x) = exp{-[1 + ξ((x-μ)/σ)]^(-1/ξ)} for ξ ≠ 0. Which sign of the shape parameter ξ corresponds to the Fréchet case, with a heavy upper tail?

The Fréchet case corresponds to a positive shape parameter ξ. Its upper tail is unbounded and decays polynomially, so it is heavy. A zero shape gives Gumbel with exponential decay, and a negative shape gives the Weibull type with a finite upper endpoint.

  1. Aξ > 0Correct
  2. Bξ = 0
  3. Cξ < 0
  4. Dξ = 1 only
  5. ξ = -1 only

Explanation

The GEV family contains three types by the sign of ξ. A positive ξ gives the Fréchet distribution, with an unbounded upper tail that decays like a power law. ξ = 0 is Gumbel, and ξ < 0 is the Weibull type with a finite upper endpoint.

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