The chapter is devoted to quick exploration techniques on how statistically decide for one of the three models, Fréchet, Gumbel or Weibull, assuming a distribution that is one of these three forms. First, it is considered a model from previous knowledge. Afterwards, the technique of the probability paper for Gumbel distribution is explored for quick statistical choice between Weibull, Gumbel and Fréchet distribution. A simple statistical choice between the extreme models is presented (trilemma). Moreover, short illustrating examples of quick exploration of data are considered.
“Assumption by tradition” essentially involves a variable mixture: a convenient usage of accumulated experimentation, of limiting results to be considered as being finitely capable of fitting data, as well as some intuition.
But what happens when the distribution underlying the sample is one of those “awkward” distributions that are not attracted to one of the max-stable distributions?
Some rules of thumb, integrating accumulated experience, can be suggested. For maxima: floods and extreme winds- Gumbel and Fréchet distributions; Largest waves- Fréchet distribution; Longest lives- Gumbel distribution. For minima: Droughts and Breaking strength of materials- Weibull distribution.
Use the Gumbel probability paper for quick statistical choice between Weibull, Gumbel and Fréchet distributions for maxima.
It seems natural to solve only the statistical trilemma (Weibull, Gumbel or Fréchet) and not a statistical dilemma or one-sided test of hypothesis of Gumbel vs. Fréchet or Gumbel vs. Weibull.