Annual rainfall maxima: Large-deviation alternative to extreme-value and extreme-excess methods

Contrary to common belief, Gumbel’s extreme value (EV) and Pickands’ extreme excess (EE) theories do not generally apply to rainfall maxima at the annual level. This is true not just for long averaging durations d, as one would expect, but also in the high-resolution limit as d->0. We reach these conclusions by studying the annual maxima of scale-invariant rainfall models with a multiplicative structure. We find that for d->0 the annual maximum rainfall intensity in d, Iyear(d), has a generalized extreme value (GEV) distribution with a shape parameter k that is significantly higher than that predicted by Gumbel’s theory and is always in the EV2 range. Under the same conditions, the excess above levels close to the annual maximum has generalized Pareto (GP) distribution with a parameter k that is always higher than that predicted by Pickands’ theory. The proper tool to obtain these results is large deviation (LD) theory, a branch of probability that has been largely ignored in stochastic hydrology.