%0 Book %1 Embrechts.etal1997 %A Embrechts, Paul %A Klüppelberg, Claudia %A Mikosch, Thomas %B Applications of Mathematics (New York) %C Berlin %D 1997 %I Springer-Verlag %K EVT univariate %P xvi+645 %T Modelling extremal events %V 33 %@ 3-540-60931-8

@book{Embrechts.etal1997, added-at = {2010-11-26T10:57:09.000+0100}, address = {Berlin}, author = {Embrechts, Paul and Klüppelberg, Claudia and Mikosch, Thomas}, biburl = {https://www.bibsonomy.org/bibtex/2af1bdb16f337be39cd46ca3fd6c21882/marsianus}, description = {MR: Publications results for "Author=(Embrechts) AND Author=(Mikosch)"}, interhash = {975a292f55e65da11f6bae8de50fb808}, intrahash = {af1bdb16f337be39cd46ca3fd6c21882}, isbn = {3-540-60931-8}, keywords = {EVT univariate}, mrclass = {60G70 (60Fxx 60K05 62M10 62P05)}, mrnumber = {1458613 (98k:60080)}, mrreviewer = {R. A. Maller}, note = {For insurance and finance}, pages = {xvi+645}, publisher = {Springer-Verlag}, review = {A reader's first impression on leafing through this book is of the large number of graphs and diagrams, used to illustrate shapes of distributions, to plot sample paths of various processes and to show real data examples in various ways. A closer reading reveals a nice mix of theory and applications, with the copious graphical illustrations alluded to. Such a mixture is of course dear to the heart of the applied probabilist/statistician, and should impress even the most ardent theorists with the range of applications of the subject. While there are a number of books available which cover most of the topics herein, I know of none which presents such a range of theory and applications of extremal processes in one volume, at a level easily understood by users of the methodology. I highly recommend the book to all who work in the area, or in related areas. The book is subtitled "For insurance and finance'', and it is well motivated in these terms. Chapter 1 covers risk theory and the "ruin problem'', based on the Cramér-Lundberg renewal model. They give, in particular, the Cramér-Lundberg theorem, which estimates the asymptotic behaviour of $\psi(u)$ψ(u) (for large $u$u) for relatively "light-tailed'' distributions, and an extension for relatively "heavy-tailed'' distributions, using regularly varying and sub-exponential assumptions on the tail behaviour of the claim size distribution. Again, a nice feature is the plotting of various simulated realisations of $U(t)$U(t) for a range of scenarios. In view of Chapter 1's setup, clear financial motivation, it is clear that a study of random walks is needed, and Chapter 2 covers this ground. Restricting themselves mainly to sums of i.i.d. random variables, the authors cover the weak and strong laws of large numbers, the law of the iterated logarithm, convergence to stable laws (including of course the central limit theorem), the Berry-Esseen theorem, and some large deviations results. Proofs of these classical results are omitted, for the most part, with reference made to standard texts. This chapter also contains a treatment of Brownian motion, and Donsker's invariance principle and its uses via the continuous mapping theorem. A section on random time changes introduces the Poisson and other point and counting processes, relating the results back to the Cramér-Lundberg model, for example. A version of Anscombe's theorem and a diffusion approximation for the risk process conclude Chapter 2. Chapter 3 covers classical extreme value theory, being concerned with the limiting behaviour of maxima of i.i.d. sequences. The usual three limit types appear through the Fisher-Tippett theorem, and their domains of attraction are treated in detail. Many examples and illustrations make this an easy development to follow. This chapter also contains some almost sure limiting results for maxima. Chapter 4 embarks on a more intensive analysis of the large sample behaviour of the "upper order statistics'' of an i.i.d. sample, and Chapter 5 continues this study by introducing the point process approach largely developed by Resnick in his publications and book [S. I. Resnick, Extreme values, regular variation, and point processes, Springer, New York, 1987; MR0900810 (89b:60241)], which gives such a deep insight into the properties of extremes and exceedances. Record values and times of an i.i.d. sequence as well as linear processes are studied here, too. Statistical methods, per se, make their appearance in Chapter 6, where the question of how extreme values manifest themselves in real data is addressed. Examples in flood and insurance data are used to motivate the discussion, Q-Q plots of data simulated from various standard distributions are shown and the "mean excess function'' (expected excess over a threshold value) is introduced as a graphical tool for assessment of the tail behaviour of the underlying distribution. The methods are thoroughly demonstrated on a couple of insurance and finance examples. Some discussion of the return period (waiting time between specific extremal events) and of the ratio of the maximum value of i.i.d. random variables to their sum, as a "heavy tail'' detection device, is included. This chapter also contains a detailed discussion of parameter estimation for extreme value distributions. The method suggested is maximum likelihood, and it is well exemplified for the Gumbel and other extreme value distributions. Hill's estimator for the index $\alpha$α of regular variation of the tail of a distribution in the domain of attraction of an extreme value distribution, and a variant due to Deckers, Einmahl and de Haan, are discussed as well. Chapter 7, a relatively short one, reviews some classical time series theory, then discusses the estimation of the autocorrelation function and the power transfer function for heavy-tailed distributions. The final chapter ("Special topics'') is large and covers a wide range of material, including the "extremal index'' which is defined for a strictly stationary sequence $X_n$.Such a $\theta$θ, the "extremal index'', if it exists, must lie in $[0,1]$[0,1], and measures to some extent the way any dependence in the data affects the behaviour of its extremes. The extremal index of a stationary sequence does not always exist, but when it does, a value of $\theta$θ near 1 typifies independence or weak dependence. Smaller values of $\theta$θ tend to indicate various kinds of dependence or clustering in the data. Section 8.2 of this chapter deals further with the relationship of a sum of i.i.d. random variables to their maximum, in the context of a "large-claim'' problem in insurance data. Section 8.4 discusses "perpetuities'', an important application in finance and insurance areas, which are modelled by a certain class of random recurrence equations which also find application in many other areas. The discussion here includes mention of classic results of Vervaat and Kesten in this area, and more recent techniques of Embrechts and Goldie. A related class of models, the ARCH (auto-regressive conditionally heteroscedastic) processes, are also analysed in this section. Chapter 8 also includes some analysis of "longest success runs'' in coin tossing, "re-insurance treaties'' (a method by which insurers spread the risk of catastrophic claims) and, finally, integrals with respect to stable processes as models for time series. Some discussion of fractional processes, including fractional Brownian motion, is included here too. The combination of skills and expertise of the three authors of this book is impressive. Their reading covers not only the traditional and classical works in the area but a great deal of the modern development, too. (They give 646 references to books and articles in the literature.) Their book concludes with copious appendices setting out the basic probability theory and some of the regular variation theory required for understanding the rest of the development. In summary, this is a worthwhile book in an extremely important area.}, series = {Applications of Mathematics (New York)}, timestamp = {2013-08-21T15:57:09.000+0200}, title = {Modelling extremal events}, volume = 33, year = 1997 }