Extreme value analysis of epoch maxima-convergence, and choice of asymptote

Cramerʹs methodology is used to derive the Type I and Type III extreme value asymptotes. The method demonstrates that both methods involve a common limiting process followed in each case by an appropriate approximation of the parent. This highlights the importance of convergence if good predictions of design values are to be obtained. It is shown that the Type III asymptote requires not just an upper limit but quite stringent mathematical conditions to hold at that limit. This implies the need to demonstrate that the underlying physics exist to create such a limit. An alternative class of right-limited distributions exists which have the Type I asymptote, and are equally if not more plausible models for phenomena right-limited by a physical effect. This class cannot be distinguished from Type III unless the parent is known. Poorly converged Type I examples are also indistinguishable from Type III, if the observations are well below the upper limit. The conclusion is that pursuit of the Type III option should be abandoned. Ensuring proper pre-conditioning before fitting the Type I asymptote offers better prospects for fitting observed extremes.