Fundamental Generation Systems

Appendix 10Asymptotic Distributions

The sample size grows very large and tends to infinity, and the distribution of the largest or smallest values may tend asymptotically to a mathematical distribution function in some cases when the samples are identically distributed and statistically independent. Some of these asymptotic distributions have a wide range of applications.

El Hami and Radi (2013b) listed three types of so-called asymptotic extreme value distributions as type I, type II and type III. Type I extreme value distributions are used extensively in mechanical reliability applications.

The distribution of maxima in sets of samples drawn from a population, following a normal distribution, converges asymptotically to this distribution. This type of distribution is used to model climatic phenomena such as wind or water level variations.

Type II distributions are also used to create models of environmental phenomena, such as earthquakes, and can be derived from the sample sets of a lognormal distribution.

Type III distributions, also known as Weibull distributions in the case of smaller values, can be obtained by the convergence of most of the distributions that have a lower bound. It is often used to describe the mechanical strength of a material or the good operating time of an electronic or mechanical device.

Extreme value distributions can be treated in the same way as other distributions, that is, defined in terms of their probability density function or distribution ...