The urgent need to describe and to solve certain problems connected to extreme phenomena in various areas of applications has been of decisive influence on the vital development of extreme value theory. After the pioneering work of M. Frechet (1927) and of R.A. Fisher and L.R.C. Tippett (1928), who discovered the limiting distributions of extremes, the importance of mathematical concepts of extreme behavior in applications was impressively demonstrated by statisticians like E.J. Gumbel and W. Weibull. The predominant role of applied aspects in that early period may be highlighted by the fact that two of the "Fisher-Tippett asymptotes" also carry the names of Gumbel and Weibull. In the last years, the complexity of problems and their tractability by mathematical methods stimulated a rapid development of mathematical theory that substantially helped to improve our understanding of extreme behavior. Due to the depth and richness of mathematical ideas, extreme value theory has become more and more of interest for mathematically oriented research workers. This was one of the reasons to organize a conference on extreme value theory which was held at the Mathematische Forschungsinstitut at Oberwolfach (FRG) in December 1987.

Klappentext

The theory of extreme values has become a broad subject which is difficult to cover by a few authors. It is the purpose of this book to lay out in an expository way the broad spectrum of extremes by contributions, some of which are reviewing recent developments and some are including original ideas and results. In the last years, the complexity of problems and their tractability by mathematical methods stimulated a rapid development of mathematical theory that substantially helped to improve our understanding of extreme behavior. Due to the depth and richness of mathematical ideas, extreme value theory has become more and more interesting to mathematically oriented research workers. This was one of the reasons a conference on extreme value theory was held at the Mathematische Forschungsinstitut at Oberwolfach (FRG) in December 1987. The book is split into three parts with a total number of nine sections. The topics covered include probabilistic theory, statistical theory of extreme values, and multivariate extremes and records. TOC:Contents: Univariate Extremes: Probability Theory: Limit Laws and Expansions. Strong Laws. Records. Exceedances. Characterizations.- Univariate Extremes: Statistics: Estimation. Test Procedures. Sufficiency of Extremes in Parametric Models.- Multivariate Extremes.- Author Index.- Subject Index.

Inhalt
I. Univariate Extremes: Probability Theory.- 1. Limit Laws and Expansions.- Best attainable rate of joint convergence of extremes.- Recent results on asymptotic expansions in extreme value theory.- 2. Strong Laws.- Strong laws for the k-th order statistic when k ? c • log2n (II).- Extreme values with heavy tails.- 3. Records.- A survey on strong approximation techniques in connection with records.- Self-similar random measures, their carrying dimension, and application to records.- 4. Exceedances.- On exceedance point processes for stationary sequences under mild oscillation restrictions.- A central limit theorem for extreme sojourn time of stationary Gaussian processes.- On the distribution of random waves and cycles.- 5. Characterizations.- Characterizations of the exponential distribution by failure rate- and moment properties of order statistics.- A characterization of the uniform distribution via maximum likelihood estimation of its location parameter.- II. Univariate Extremes: Statistics.- 1. Estimation.- Simple estimators of the endpoint of a distribution.- Asymptotic normality of Hill's estimator.- Extended extreme value models and adaptive estimation of the tail index.- Asymptotic results for an extreme value estimator of the autocorrelation coefficient for a first order autoregressive sequence.- 2. Test Procedures.- The selection of the domain of attraction of an extreme value distribution from a set of data.- Comparison of extremal models through statistical choice in multidimensional backgrounds.- 3. Sufficiency of Extremes in Parametric Models.- The role of extreme order statistics for exponential families.- III. Multivariate Extremes.- Multivariate records and shape.- Limit distributions of multivariate extreme values in nonstationary sequences ofrandom vectors.- Statistical decision for bivariate extremes.- Multivariate negative exponential and extreme value distributions.- Author Index.