Dispersion Models for Extremes

We propose extreme value analogues of natural exponential families and exponential dispersionmodels and introduce the hazard slope as an analogue of the variance function. The set of quadraticand power hazard slopes is characterized, and is shown to include well-known distributions such asthe Rayleigh, Gumbel, power, Pareto, logistic, negative exponential, Weibull and Fréchet families. Weshow a convergence theorem for hazard slopes, which gives rise to a new version of the classical extremevalue convergence results in the extreme dispersion model setting. The main idea of the paper is toexplore the parallels between location families and natural exponential families, and between convolutionand the minimum operation.