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2011 | Buch

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Laws of Small Numbers: Extremes and Rare Events

verfasst von: Michael Falk, Jürg Hüsler, Rolf-Dieter Reiss

Verlag: Springer Basel

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

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

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Inhaltsverzeichnis

Frontmatter

The IID Case: Functional Laws of Small Numbers

Frontmatter

Chapter 1. Functional Laws of Small Numbers

Abstract

We will develop in the following a particular extension of the well-known Poisson approximation of binomial distributions with a small hitting probability, which is known as the law of small numbers. This extension, which one might call functional laws of small numbers, links such seemingly different topics like non-parametric regression analysis and extreme value theory.

Michael Falk, Jürg Hüsler, Rolf-Dieter Reiss

Chapter 2. Extreme Value Theory

Abstract

In this chapter we summarize results in extreme value theory, which are primarily based on the condition that the upper tail of the underlying df is in the δ-neighborhood of a generalized Pareto distribution (GPD). This condition, which looks a bit restrictive at first sight (see Section 2.2), is however essentially equivalent to the condition that rates of convergence in extreme value theory are at least of algebraic order (see Theorem 2.2.5). The δ-neighborhood is therefore a natural candidate to be considered, if one is interested in reasonable rates of convergence of the functional laws of small numbers in extreme value theory (Theorem 2.3.2) as well as of parameter estimators (Theorems 2.4.4, 2.4.5 and 2.5.4).

Michael Falk, Jürg Hüsler, Rolf-Dieter Reiss

Chapter 3. Estimation of Conditional Curves

Abstract

In this chapter we will pick up Example 1.3.3 again, and we will show how the Poisson approximation of truncated empirical point processes enables us to reduce conditional statistical problems to unconditional ones.

A nearest neighbor alternative to this applications of our functional laws of small numbers is given in Sections 3.5 and 3.6.

Michael Falk, Jürg Hüsler, Rolf-Dieter Reiss

The IID Case: Multivariate Extremes

Frontmatter

Chapter 4. Basic Theory of Multivariate Maxima

Abstract

In this chapter, we study the limiting distributions of componentwise defined maxima of iid d-variate rv. Such distributions are again max-stable as in the univariate case. Some technical results and first examples of max-stable df are collected in Section 4.1. In Section 4.2 and 4.3, we describe representations of max-stable df such as the de Haan-Resnick and the Pickands representation. Of special interest for the subsequent chapters will be the Pickands dependence function in Section 4.3 and the D-norm, which will be introduced in Section 4.4.

Michael Falk, Jürg Hüsler, Rolf-Dieter Reiss

Chapter 5. Multivariate Generalized Pareto Distributions

Abstract

In analogy to the univariate case, we introduce certain multivariate generalized Pareto df (GPD) of the form W = 1 + log(G) for the statistical modelling of multivariate exceedances, see Section 5.1. Various results around the multivariate peaks-over-threshold approach are compiled in Section 5.2. The peaks-overthreshold stability of a multivariate GPD is investigated in Section 5.3.

Michael Falk, Jürg Hüsler, Rolf-Dieter Reiss

Chapter 6. The Pickands Approach in the Bivariate Case

Abstract

The restriction to bivariate rv enables the study of their distributions in much greater detail. We introduce, for example, a certain measure generating function M, see Section 6.1, and prove that the pertaining Pickands dependence function Dis absolutely continuous, see Lemma 6.2.1 and the subsequent discussion. This property is unknown in higher dimensions.

Michael Falk, Jürg Hüsler, Rolf-Dieter Reiss

Chapter 7. Multivariate Extremes: Supplementary Concepts and Results

Abstract

In this chapter we will deal with exceedances and upper order statistics (besides maxima), with the point process approach being central for these investigations. Extremes will be asymptotically represented by means of Poisson processes with intensity measures given by max-Lévy measures as introduced in Section 4.3.

Michael Falk, Jürg Hüsler, Rolf-Dieter Reiss

Non-IID Observations

Frontmatter

Chapter 8. Introduction to the Non-IID Case

Abstract

We present in the following some examples to motivate the extension of the classical extreme value theory for iid sequences to a theory for non-iid sequences. We introduce different classes of non-iid sequences together with the main ideas. The examples show that suitable restrictions for each class are needed to find limit results which are useful for applications.

Michael Falk, Jürg Hüsler, Rolf-Dieter Reiss

Chapter 9. Extremes of Random Sequences

Abstract

We develop the general theory of extremes and exceedances of high boundaries by non-stationary random sequences. Of main interest is the asymptotic convergence of the point processes of exceedances or of clusters of exceedances. These results are then applied for special cases, as stationary, independent and particular nonstationary random sequences.

Michael Falk, Jürg Hüsler, Rolf-Dieter Reiss

Chapter 10. Extremes of Gaussian Processes

Abstract

In this chapter continuous Gaussian processes and their extremes, exceedances and sojourns above a boundary are treated. Results are derived for stationary and locally stationary Gaussian processes. The asymptotic results are then applied to a statistical problem related to empirical characteristic functions. In addition, some results on other non-stationary Gaussian processes are discussed. The relation between the continuous process and its discrete approximation on a certain fine grid is a rather interesting issue, in particular for simulations or approximations.

Michael Falk, Jürg Hüsler, Rolf-Dieter Reiss

Chapter 11. Extensions for Rare Events

Abstract

In the following sections we discuss some extensions which were mentioned in the previous chapters. Of main interest is the extension to general rare events in relation to a random sequence applying the same method as used for dealing with exceedances. In addition we treat now also the point process of all exceedances if clustering occurs. These results are applied to the processes of peaks over a threshold and of rare events. Finally, in the same way general rare events are considered without relation to a random sequence. Here triangular arrays of rare events will be analyzed by the same approach. This extension unifies easily the different local dependence conditions. Its application to multivariate extremes is then straightforward. As a particular case, triangular arrays of rare events in relation with exceedances of Gaussian sequences are considered since they are basic for maxima of a continuous Gaussian process. This analysis reveals also a new definition of Pickands constants.

Michael Falk, Jürg Hüsler, Rolf-Dieter Reiss

Chapter 12. Statistics of Extremes

Abstract

We use in the following the theory developed in the preceding chapters to discuss a few nonstandard applications. Of interest are here the statistical estimation of the cluster distribution and of the extremal index in a stationary situation. In the last section we treat a frost data problem which is related to an extreme value problem of a nonstationary sequence.

Michael Falk, Jürg Hüsler, Rolf-Dieter Reiss

Backmatter

Titel: Laws of Small Numbers: Extremes and Rare Events
verfasst von: Michael Falk
Jürg Hüsler
Rolf-Dieter Reiss
Copyright-Jahr: 2011
Verlag: Springer Basel
Electronic ISBN: 978-3-0348-0009-9
Print ISBN: 978-3-0348-0008-2
DOI: https://doi.org/10.1007/978-3-0348-0009-9