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

Block maxima of a loss series are modelled by a Generalised Extreme Value (GEV) distribution with shape parameter xi. Which statement correctly links the sign of xi to the tail and the Fréchet, Gumbel and Weibull families?

A positive shape parameter gives the Fréchet type of GEV, which has a heavy, power-law tail and no finite upper endpoint. Zero gives Gumbel with an exponential-type tail, and a negative value gives Weibull with a bounded upper end.

  1. Axi > 0 gives the Fréchet type, with a heavy tail and no finite upper endpointCorrect
  2. Bxi > 0 gives the Weibull type, with a finite upper endpoint
  3. Cxi = 0 gives the Fréchet type, with a polynomially decaying tail
  4. Dxi < 0 gives the Fréchet type, with a heavy tail
  5. xi < 0 gives the Gumbel type, with an exponentially decaying tail

Explanation

In the GEV family, xi > 0 is Fréchet (heavy, power-law tail, unbounded above), xi = 0 is Gumbel (exponential-type tail) and xi < 0 is Weibull (finite upper endpoint). The Weibull option fails because it attaches the Weibull type to xi > 0 when it belongs to xi < 0.

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