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2001 | Buch

Introduction Kapitel lesen Erstes Kapitel lesen

An Introduction to Statistical Modeling of Extreme Values

verfasst von: Stuart Coles

Verlag: Springer London

Buchreihe : Springer Series in Statistics

Enthalten in: Professional Book Archive

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Über dieses Buch

Directly oriented towards real practical application, this book develops both the basic theoretical framework of extreme value models and the statistical inferential techniques for using these models in practice. Intended for statisticians and non-statisticians alike, the theoretical treatment is elementary, with heuristics often replacing detailed mathematical proof. Most aspects of extreme modeling techniques are covered, including historical techniques (still widely used) and contemporary techniques based on point process models. A wide range of worked examples, using genuine datasets, illustrate the various modeling procedures and a concluding chapter provides a brief introduction to a number of more advanced topics, including Bayesian inference and spatial extremes. All the computations are carried out using S-PLUS, and the corresponding datasets and functions are available via the Internet for readers to recreate examples for themselves. An essential reference for students and researchers in statistics and disciplines such as engineering, finance and environmental science, this book will also appeal to practitioners looking for practical help in solving real problems. Stuart Coles is Reader in Statistics at the University of Bristol, UK, having previously lectured at the universities of Nottingham and Lancaster. In 1992 he was the first recipient of the Royal Statistical Society's research prize. He has published widely in the statistical literature, principally in the area of extreme value modeling.

Inhaltsverzeichnis

Frontmatter
1. Introduction
Abstract
Extreme value theory has emerged as one of the most important statistical disciplines for the applied sciences over the last 50 years. Extreme value techniques are also becoming widely used in many other disciplines. For example: for portfolio adjustment in the insurance industry; for risk assessment on financial markets; and for traffic prediction in telecommunications. At the time of writing, in the past twelve months alone, applications of extreme value modeling have been published in the fields of alloy strength prediction (Tryon & Cruse, 2000); ocean wave modeling (Dawson, 2000); memory cell failure (McNulty et al., 2000); wind engineering (Harris, 2001); management strategy (Dahan & Mendelson, 2001); biomedical data processing (Roberts, 2000); thermodynamics of earthquakes (Lavenda & Cipollone, 2000); assessment of meteorological change (Thompson et al., 2001); non-linear beam vibrations (Dunne & Ghanbari, 2001); and food science (Kawas & Moreira, 2001).
Stuart Coles
2. Basics of Statistical Modeling
Abstract
It is easiest to introduce concepts by way of example. Suppose we are interested in studying variations from day to day in rainfall levels measured at a particular location. The sequence of observed daily rainfall levels constitute the data, denoted x 1,...x n On any particular day, prior to measurement, the rainfall level is an uncertain quantity: even with sophisticated weather maps, future rainfall levels cannot be predicted exactly. So, the rainfall on day i is a random quantity, X i . Once measured, the value is known to be x i . The distinction between lower and upper case letters is that the upper-case X i represents the random quantity, whose realized value is subsequently measured as the lower-case x i . Obviously, although X i is a random quantity, in the sense that until measured it could take a range of different values, some values are more likely than others. Thus, X i is assumed to have a probability distribution which attaches probabilities to the various values or ranges of values that X i might take, and values that are more likely have a higher probability than those which are not.
Stuart Coles
3. Classical Extreme Value Theory and Models
Abstract
In this chapter we develop the model which represents the cornerstone of extreme value theory.
Stuart Coles
4. Threshold Models
Abstract
As discussed in Chapter 3, modeling only block maxima is a wasteful approach to extreme value analysis if other data on extremes are available. Though the r largest order statistic model is a better alternative, it is unusual to have data of this form. Moreover, even this method can be wasteful of data if one block happens to contain more extreme events than another. If an entire time series of, say, hourly or daily observations is available, then better use is made of the data by avoiding altogether the procedure of blocking.
Stuart Coles
5. Extremes of Dependent Sequences
Abstract
Each of the extreme value models derived so far has been obtained through mathematical arguments that assume an underlying process consisting of a sequence of independent random variables. However, for the types of data to which extreme value models are commonly applied, temporal independence is usually an unrealistic assumption. In particular, extreme conditions often persist over several consecutive observations, bringing into question the appropriateness of models such as the GEV. A detailed investigation of this question requires a mathematical treatment at a greater level of sophistication than we have adopted so far. However, the basic ideas are not difficult and the main result has a simple heuristic interpretation. A more precise development is given by Leadbetter et al. (1983).
Stuart Coles
6. Extremes of Non-stationary Sequences
Abstract
Non-stationary processes have characteristics that change systematically thorough time. In the context of environmental processes, non-stationarity is often apparent because of seasonal effects, perhaps due to different climate patterns in different months, or in the form of trends, possibly due to long-term climate changes. Like the presence of temporal dependence, such departures from the simple assumptions that were made in the derivation of the extreme value characterizations in Chapters 3 and 4 challenge the utility of the standard models. In Chapter 5 we were able to demonstrate that, in a certain sense and subject to specified limitations, the usual extreme value limit models are still applicable in the presence of temporal dependence. No such general theory can be established for non-stationary processes. Results are available for some very specialized forms of non-stationarity, but these are generally too restrictive to be of use for describing the patterns of non-stationarity found in real processes. Instead, it is usual to adopt a pragmatic approach of using the standard extreme value models as basic templates that can be enhanced by statistical modeling.
Stuart Coles
7. A Point Process Characterization of Extremes
Abstract
There are different ways of characterizing the extreme value behavior of a process, and a particularly elegant formulation is derived from the theory of point processes. The mathematics required for a formal treatment of this theory is outside the scope of this book, but we can again give a more informal development. This requires just basic ideas from point process theory. In a sense, the point process characterization leads to nothing new in terms of statistical models; all inferences made using the point process methodology could equally be obtained using an appropriate model from earlier chapters. However, there are two good reasons for considering this approach. First, it provides an interpretation of extreme value behavior that unifies all the models introduced so far; second, the model leads directly to a likelihood that enables a more natural formulation of non-stationarity in threshold excesses than was obtained from the generalized Pareto model discussed in Chapters 4 and 6.
Stuart Coles
8. Multivariate Extremes
Abstract
In Chapters 3 to 7 we focused on representations and modeling techniques for extremes of a single process. We now turn attention to multivariate extremes. When studying the extremes of two or more processes, each individual process can be modeled using univariate techniques, but there are strong arguments for also studying the extreme value inter-relationships. First, it may be that some combination of the processes is of greater interest than the individual processes themselves; second, in a multivariate model, there is the potential for data on each variable to inform inferences on each of the others. Examples 1.9–1.11 illustrate situations where such techniques may be applicable.
Stuart Coles
9. Further Topics
Abstract
In Chapter 3 we discussed a number of different techniques for parameter estimation in extreme value models and argued that likelihood-based methods are preferable. Subsequently, all our analyses have adopted the procedure of maximum likelihood. But this is not the only way to draw inferences from the likelihood function, and Bayesian techniques offer an alternative that is often preferable.
Stuart Coles
Backmatter
Metadaten
Titel
An Introduction to Statistical Modeling of Extreme Values
verfasst von
Stuart Coles
Copyright-Jahr
2001
Verlag
Springer London
Electronic ISBN
978-1-4471-3675-0
Print ISBN
978-1-84996-874-4
DOI
https://doi.org/10.1007/978-1-4471-3675-0