A new type of extreme value distributions

The three classical pairs of extreme value distributions correspond to random variables with ranges of values unbounded from either one or both sides. Some applications of statistics of extremes, however, deal with variables, which are bounded on both sides (local values of specific fracture energy in crack diffusion theory is one such example). In this paper, we derive a fourth pair of extreme value distributions, which are supported on a finite segment (one for maxima and one for minima). First, we propose a derivation of the three known maximal value distributions, which lends itself to a generalization (everything is done for maxima, since transition to minima is standard). The derivation is then extended to a slightly more general setting, and the fourth distribution is obtained. It is explained that certain fact concerning groups of transformations of the real line prevents any further generalization, i.e. the extended list of extreme value distributions is complete. The three classical maximal value distributions can be obtained as limits of the new one. A possible criterion of when one may expect the new distribution to be more adequate than the Weibull distribution is offered. An illustrative numerical example is considered, in which the scatter of sample minima is modeled by both Weibull and the new distribution. Another example shows that when the modeling of data requires very high values of the shape parameter of the Weibull distribution, the new distribution may be expected to have much smaller "shape parameter" values. The modeling of experimentally observed scatter of crack arrest length, using the Weibull distribution, is compared to that using the new distribution. (C) 1997 Elsevier Science Ltd.