Classical Extreme Value Theory-the asymptotic distributional theory for maxima of independent, identically distributed random variables-may be regarded as roughly half a century old, even though its roots reach further back into mathematical antiquity. During this period of time it has found significant application-exemplified best perhaps by the book Statistics of Extremes by E. J. Gumbel-as well as a rather complete theoretical development. More recently, beginning with the work of G. S. Watson, S. M. Berman, R. M. Loynes, and H. Cramer, there has been a developing interest in the extension of the theory to include, first, dependent sequences and then continuous parameter stationary processes. The early activity proceeded in two directions-the extension of general theory to certain dependent sequences (e.g., Watson and Loynes), and the beginning of a detailed theory for stationary sequences (Berman) and continuous parameter processes (Cramer) in the normal case. In recent years both lines of development have been actively pursued.
Since the publication of the first edition of this seminar book in 1994, the theory and applications of extremes and rare events have enjoyed an enormous and still increasing interest. The intention of the book is to give a mathematically oriented development of the theory of rare events underlying various applications. This characteristic of the book was strengthened in the second edition by incorporating various new results. In this third edition, the dramatic change of focus of extreme value theory has been taken into account: from concentrating on maxima of observations it has shifted to large observations, defined as exceedances over high thresholds. One emphasis of the present third edition lies on multivariate generalized Pareto distributions, their representations, properties such as their peaks-over-threshold stability, simulation, testing and estimation. Reviews of the 2nd edition: "In brief, it is clear that this will surely be a valuable resource for anyone involved in, or seeking to master, the more mathematical features of this field." David Stirzaker, Bulletin of the London Mathematical Society "Laws of Small Numbers can be highly recommended to everyone who is looking for a smooth introduction to Poisson approximations in EVT and other fields of probability theory and statistics. In particular, it offers an interesting view on multivariate EVT and on EVT for non-iid observations, which is not presented in a similar way in any other textbook." Holger Drees, Metrika
This volume consists of 22 research papers by leading researchers in Probability and Statistics. Many of the papers are focused on themes that Professor Bhattacharya has published on research. Topics of special interest include nonparametric inference, nonparametric curve fitting, linear model theory, Bayesian nonparametrics, change point problems, time series analysis and asymptotic theory. This volume presents state-of-the-art research in statistical theory, with an emphasis on nonparametric inference, linear model theory, time series analysis and asymptotic theory. It will serve as a valuable reference to the statistics research community as well as to practitioners who utilize methodology in these areas of emphasis.
This comprehensive text gives an interesting and useful blend of the mathematical, probabilistic and statistical tools used in heavy-tail analysis. Heavy tails are characteristic of phenomena where there is a significant probability of a single huge value impacting system behavior. Record-breaking insurance losses, financial returns, sizes of files stored on a server and transmission rates of files are all examples of heavy-tailed phenomena. Prerequisites for the reader include a prior course in stochastic processes and probability, some statistical background, some familiarity with time series analysis, and ability to use (or at least to learn) a statistics package such as R or Splus.
J. Neyman, one of the pioneers in laying the foundations of modern statistical theory, stressed the importance of stochastic processes in a paper written in 1960 in the following terms: Currently in the period of dynamic indeterminism in science, there is hardly a serious piece of research, if treated realistically, does not involve operations on stochastic processes. Arising from the need to solve practical problems, several major advances have taken place in the theory of stochastic processes and their applications. Books by Doob (1953; J. Wiley and Sons), Feller (1957, 1966; J. Wiley and Sons) and Loeve (1960; D. van Nostrand and Col., Inc.) among others, have created growing awareness and interest in the use of stochastic processes in scientific and technological studies.The literature on stochastic processes is very extensive and is distributed in several books and journals.