Laws of Small Numbers: Extremes and Rare Events
Extremes and Rare Events
Published in 1994
Book
Paperback/Softback
XII, 282 pages
978-3-7643-5071-0 (ISBN)
Description
Laws of small numbers concern rare events, i.e., events having but a small probability of occurrence. It is reasonable to describe such events as truncated empirical point processes, and a point process of rare events can be approximated by a Poisson process. This approach is applied to the following topics: extremes and rare events of stochastic processes, with special emphasis given to Gaussian processes; Poisson point processes; statistics in extreme value models; and nonparametric regression analysis. This book comes with the statistical software system XTREMES, version 1.2, an interactive menu-driven system which runs on IBM-compatible PCs under MS-DOS and is designed for dealing with various questions in extreme value analysis. The beginner should appreciate the introduction to the fields mentioned above, and the expert the comprehensive account of recent results and developments.
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Content
Part 1 The IID case - functional laws of small numbers: functional laws of small numbers - bounds for the functional laws of small numbers, applications; extreme value theory - Von Mises conditions, the peaks over threshold method, initial estimation of the class index; estimation of conditional curves - Poisson process approach, the nonparametric case, the semiparametric case, extension to several points, a nearest neighbour alternative, optimal accuracy of estimators; multivariate maxima - limiting distributions, representations and dependence functions, Max-Stable stochastic processes; multivariate extremes - strong approximation of exceedances, further concepts of extremes, extreme value analysis with XTREMES - artificial data, real grouped data, real continuous data. Part 2 Non IID observations: introduction to the non IID case - definitions, stationary random sequences, independent random sequences, nonstationary random sequences; extremes of random sequences - introduction and general theory, stationary sequences, independent sequences, nonstationary sequences; extremes of Gaussian processes - stationary Gaussian processes, nonstationary Gaussian processes, empirial characteristic functions; extensions for rare events - rare events of random sequences, the point process of exceedances, application to peaks over threshold, application to rare events, triangular arrays of rare events, multivariate extremes of non IID sequences; statistics of extremes - application to ecological data, frost data; appendix - user's guide to XTREMES - getting started, preparations, becoming acquainted with XTREMES, hot keys, window options and help menus; XTREMES in action - implemented distributions, generating and reading data, nonparametric estimation, estimation in the alpha-mode, estimation in the beta-mode, simulations, important facilities and advice.