  We propose extreme value analogues of natural exponential families and
exponential dispersion models, and introduce the slope function as an analogue
of the variance function. The set of quadratic and power slope functions
characterize well-known families such as the Rayleigh, Gumbel, power, Pareto,
logistic, negative exponential, Weibull and Fr\'echet. We show a convergence
theorem for slope functions, by which we may express the classical extreme
value convergence results in terms of asymptotics for extreme dispersion
models. The main idea is to explore the parallels between location families and
natural exponential families, and between the convolution and minimum
operations.
