Extreme‐Value Distributions

Abstract

This is an expository survey of the (univariate) theory of extreme values and the estimation of the parameters of the related extremevalue distributions. This survey was written at the invitation of Professors N.L. Johnson and S. Kotz, editors of the forthcoming Encyclopedia of Statiatical Seneee'to be published by John Wiley and Sons, Inc., New York. The theory of extreme values plays a fundamental role in several areas of applied statistics, such as the analysis of flood flows, the reliability of complex systems, the analysis of air pollution data, etc. In addition to surveying (without proofs) the basic results of extreme value theory and the estimators of the parameters of the extreme value distributions, this survey presents a brief discussion of the current research in these areas. The theory of extreme values, and the extreme value distributions, play an important role in theoretical and applied statistics. For example, extreme value distributions arise quite naturally in the study of size effect on material strengths, the occurrence of floods and droughts, the reliability of systems made up of a large number of

Spatial dependences among precipitation maxima over Belgium

“…Consequently, we need to check the validity of this assumption for our selected stations. The Mann test (Mann, 1945) which provides information on the presence of tendencies has been applied to the ensemble of maxima time series (for a duration of 1 day). The analysis reveals that the vast majority of the time series recorded for summer are stationary.…”

“…For example, applied and theoretical statisticians (e.g. Capéraà et al, 1997;de Haan and Pereira, 2006;Hall and Tajvidi, 2000;Pickands, 1981), have proposed and explored many directions to infer the characteristics of the bivariate structure between maxima. Several parametric families have also been studied (e.g.…”

Methods for obtaining true particle size distributions from cross section measurements

“…True Diameters, r or Measured Diameters, x True Population Density, F(r) or Probability of Measurement, (x) needs to be estimated using the theory of extreme values [20], which has been examined by Kötzer and Molchanov [21]. Wicksell found this technique to be useful when only the true distribution of several hundreds of samples is evaluated.…”