Extreme Value Theory Fundamentals for MAS-I

Block Maxima Approach

Extreme value theory (EVT) models the behavior of the largest (or smallest) values in a sample. The Fisher-Tippett-Gnedenko theorem states that the normalized maximum of n i.i.d. observations converges to one of three distributions: Gumbel (xi=0, thin tails), Frechet (xi>0, heavy tails), or Weibull (xi<0, bounded tails). These unify into the Generalized Extreme Value (GEV) distribution with shape parameter xi, location mu, and scale sigma. For insurance, the Frechet domain (xi>0) is most relevant because loss distributions typically have heavy tails.

Peaks Over Threshold

The peaks over threshold (POT) approach models exceedances above a high threshold u. The Pickands-Balkema-de Haan theorem states that for sufficiently high u, the excess distribution converges to the Generalized Pareto Distribution (GPD) with shape xi and scale sigma_u. The mean excess function e(u) = E[X-u | X>u] is linear in u for GPD, providing a diagnostic for threshold selection. A linear mean excess plot suggests GPD is appropriate above that threshold. Exam MAS-I covers GEV and GPD fitting, threshold selection, and tail probability estimation.