The Weibull distribution as an extreme value model for transformed annual maxima

Abstract

There is no mathematical difference between theoretical extreme value models of maxima and minima because they link to each other by a simple sign reversal transformation. However, other transformations that change sample maxima to sample minima raise the possibility of alternatives to the generalised extreme value (GEV) distribution for annual maxima, while still maintaining extreme value justification. A general class of transformation is proposed that converts positive annual maxima to lower-bounded minima, then amenable to Weibull extreme value analysis for sufficiently large sample sizes. That is, the Weibull distribution of smallest extremes provides a theoretical extreme value model for the transformed maxima, which holds irrespective of any GEV form of the original maxima. This would apply, for example, to the analysis of reciprocals of discharge or rainfall annual maxima. A useful feature of the transformation approach is that alternative prediction expressions with extreme value justification can arise for defining event magnitude as a function of return period. These expressions may result in different hydrological conclusions than from a GEV analysis of maxima. A double exponential transformation is introduced and its prediction function for maxima is noted to have capability to mimic equivalent Type 3 extreme value (EV3) distribution expressions, but without having to introduce an upper bound parameter. The new function gives a good fit to apparent EV3 annual flood maxima recorded from two very different catchments: the Yangtze River in China and the upper Whanganui River in New Zealand.