In: JASA, 93, Dec. 1998, pp. 1516-1518

... is about application of EVT and not about the probability theory underlying the models, although an idea is given of the principle results. The book's main audience will be applied statisticians who want to model data on extremes. The theorists would be advised to turn to the work of Embrechts et al. (1997), which contains a clearer mathematical development of the results and a much fuller set of references to the extreme value literature. A further theoretical text that would aid deeper understanding is that of Falk, Hüsler, and Reiss (1994), which shares an author with the present book and clearly takes a similar approach to modeling extremes. The older book was also published with a version of the Xtremes software, although little discussion of statistical issues.

For a practitioner from industry, insurance, or finance who is less concerned with understanding the underlying probability and more interested in applications, this new book will be a good buy. ...

Chapter 2 takes us back a step, discussing what we do when first confronted with data. I completely agree with the author's emphasis on visual tools such as sample distribution functions, histograms, kernel density estimates, sample excess and hazard functions, and Q-Q and P-P plots. In an extreme value analysis, a graphical investigation of the tail of the data is always the first step. A section on non-iid data and clustering possibly comes too early and could have postponed until after the material on the extremal index in Chapter 6.

Part II concerns parametric statistical inference. An entire chapter is devoted to normal and Poisson models for nonextreme features of the data. The stated intention is to give an outline of the author's approach within a familiar setting, and the chapter contains short notes on many topics in statistical inference such as likelihood estimation, M estimation, confidence intervals, the bootstrap, and hypothesis testing. An experienced statistician will be impatient to get beyond this, whereas a novice statistician may find the treatment too cursory te be beneficial. I feel that many of the refresher units in this chapter could have been assumed, although I appreciate that the book's second role as a manual to the software requires that something be said about modeling of nonextreme data, as this is also a feature of the software.

Chapters 4, 5, and 6 are the heart of the book, containing the material of most interest to most readers. These chapters address the modeling of block maxima, the modeling of exceedances, and advanced modeling topics. A wealth of models and an impressive array of fitting techniques are presented, but my general feeling is that subsequent stages of statistical inference have been neglected. There is very little on the calculation of realistic errors for estimates and their funtionals and the development of regression frameworks to deal with covariate information. Nor is there much critical comparison of models or discussion of the relative performance of fitting methods. I think that readers might appreciate some narrowing down of the choices. ...

Section III, consisting of Chapters 7 and 8, is a very brief excursion into multivariate statistics that, as the authors admit in the Preface, was not in the book's original plan. However, models for multivariate extremes have been a considerable growth area in the last few years, and in certain areas of application (e.g., finance) there is a very real need to be able to model joint extreme events, such as joint large losses in several stocks or joint large changes in several exchange rates. Multivariate EVT is more difficult than univariate EVT, and texts like those of Falk, Hüsler, and Reiss (1994) and Resnick (1987) are not always easy reading. I have not yet seen an accesible account of the theory in an applied text, and this book gives little theory.

On the practical side is the problem of defining and interpreting multivariate extremes due to the lack of natural ordering in dimensions greater than one. Generally speaking, data are again collected in two formats: component-wise block maxima of random vectors, which need not necessarily occur at the same time, and data points that exceed high thresholds in one or more marginals. These kinds of data again lead to block maxima models and threshold models, and, as in the univariate case, it is the latter models that seem to be more useful. Reiss and Thomas restrict their consideration to block maxima and describe three models for bivariate data that have been implemented in Xtremes: the Marshall-Olkin, Gumbel-McFadden, and Hüsler-Reiss models. Their approach to fitting these is a two-stage process in which the one-dimensional marginals are estimated first, and then a one-parameter copula or dependence function is estimated. In multivariate EVT there is no finite parameter familiy of limiting distributions as there is in the univariate case, although the limiting form of the marginals is clearly governed by univariate EVT. Considerable flexibility exists for choosing the form of the dependence structure, and this book only really scratches the surface of the models now available. I think some reference could have been made to work by Smith (1994) and Tawn and Coles (1991) for multivariate threshold data. ...

In summary, Statistical Modeling of Extremes is a worthwhile acquisition for the applied statistical modeler working with data on extremes. A practitioner from industry, insurance, or finance with limited statistical experience might find it particularly good. However, it is not yet the definite text on statistical inference in extreme value models and there remains a recognized need in the extremes community for a book that will fill this niche. ...

Alexander J. McNeil, Zürich