Body of results describing the asymptotic behavior of the largest observations in a sample, allowing a percentile to be estimated beyond the largest loss ever observed.
An insurer has to quote a 99.5% percentile while rarely holding more than twenty years of data: the number sought lies beyond anything it has seen. Fitting a distribution to all claims and then reading its tail is the wrong answer, because the fit is dominated by the thousands of small claims in the center and ignores the five observations that govern the tail. Extreme value theory reverses the approach: it models only the extremes, and it has two limit theorems for that, playing the role the central limit theorem plays elsewhere. The first, from Fisher, Tippett and Gnedenko, says that a properly renormalized sample maximum can only converge to a generalized extreme value distribution. The second, from Pickands, Balkema and de Haan, says that exceedances over a high threshold converge to a generalized Pareto distribution. A single parameter, the tail index, then summarizes extreme risk, and its sign decides whether the distribution has a finite upper bound or not.
Ronald Fisher and Leonard Tippett published the three types theorem in 1928, Boris Gnedenko gave the full proof in 1943, and James Pickands established the threshold exceedance result in 1975. The best known application outside insurance is the redesign of Dutch sea defenses after the February 1953 storm surge: the Delta Commission set the protection level for South Holland at a ten thousand year return period, a figure no simple count on a few decades of tide records could reach.
TVE, EVT, statistique des extrêmes, extreme value statistics