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Extreme Value Theory: An IntroductionPeter C Kiesslera

a Clemson UniversityPublished online: 01 Jan 2012.

To cite this article: Peter C Kiessler (2008) Extreme Value Theory: An Introduction, Journal of the American StatisticalAssociation, 103:482, 882-883, DOI: 10.1198/jasa.2008.s231

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882 Book Reviews

alphabetical order, so on any subset of pages a wide range of levels of mathe-matical and technical detail is provided; for example, pages 176–239 presentsbasic topics including boxplots, bubble plots, and case studies adjacent to morespecific/complex topics, including the Bradley–Terry model, the Breslow–Daystatistics, and catastrophe theory. Interspersed are two catalogs, one of paramet-ric tests and the other of probability density functions, along with smattering ofshort sketches on William Brown, Cyril Burt, Robert Bush, and Don Campbell.Obviously, this encyclopedia is not targeted at a narrow audience.

The editors hoped that the encyclopedia would meet the needs of a widereadership, ranging from those who need the most basic to those wanting to fitcomplex models, such as stochastic differential equations, and that the techni-cal level and mathematical content of particular entries would match the type ofreader likely to access it. Although it is a difficult matching problem, the editorshave done their job well. I asked several behavioral science undergraduates toread entries on topics covered in our introductory statistics courses, and theyfound them to be comprehensible and useful. I asked several graduate studentsto read entries on topics covered in our first-year graduate program, and theyalso found that most entries were written at a level matching their statisticalfluency and technical mastery.

EoSBS, like the Encyclopedia of Statistical Science (EoSS) and Encyclo-pedia of Biostatistics (EoB), is broad in its coverage, but EoSBS places moreemphasis on structural equation modeling, factor analysis, scaling, and mea-surement and less on statistical theory. Everitt and Howell argue that the threeencyclopedias are complementary rather than competitive. This may be true,but lines and coverage are far from distinct as interdisciplinary research be-comes the norm, and it becomes nearly impossible to find a statistical methodthat is used exclusively in behavioral sciences or a statistical method that is notused in behavioral sciences. Moreover, whereas the three encyclopedias maybe complementary, all three have a bigger competitor, which is free and at ourfingertips: the Internet.

How big of a competitor is the Internet? I picked several terms for compar-ison: sampling distribution, p values, statistical significance, empirical Bayes,and hierarchical model. I compared descriptions of these terms in EoSBS andInternet searches. Although I did not have the electronic version of the EoSBS,which based on the publisher’s website is now available, the Internet was fareasier to use and provided multiple entries for each topic. How did the qual-ity of coverage compare? In my small sample, EoSBS won hands down in thesense that I found no embarrassingly bad entries, whereas on the Internet thesewere fairly abundant. The Internet required more time-consuming sifting, ofwhich someone learning the topic would likely be unable to do. However, withthe ability to do this sifting, I usually found a description on the Internet onpar with what EoSBS delivered, and the sifting itself was educational and oftenpreferable to juggling between four heavy volumes. Given the fact that manylearners will not have the ability to do this sifting, or the $1,800 with which tobuy the encyclopedia, which information source wins (i.e., has greater impact intraining users of these methods) will be determined by how many departmentsand libraries make EoSBS available institutionally.

As mentioned earlier, the organization of the EoSBS is alphabetical ratherthan grouping within section topics. This is mostly okay, especially if one usesthe index rather than the table of contents. In the latter, parametric tests andprobability density functions are listed under C, because these topics are pre-sented as catalogs. Unfortunately, this organization means that one is usuallyjuggling several volumes each time one looks up a topic. For example, foranalysis of variance (ANOVA), Volume 1 has about a dozen pages of overview,but for more specific subtopics, fractional factorial design is in Volume 1, hier-archical designs are in Volumes 1 and 2, multitrait multimethod analysis is inVolume 3, and within-case designs are in Volume 4. Each time that I wantedto use the encyclopedia, I wished for an electronic copy, and often the volumes(like my umbrella) were not in the location at which I needed them.

Another possible improvement in organization of the encyclopedia wouldbe to eliminate redundancy in coverage of topics, especially when the redun-dant coverage uses different notation. Again, I use ANOVA as an example.Some general basic coverage presented in the Analysis of Variance section isrepeated in the ANOVA subsection of the Catalogue of Parametric Tests sec-tion, but different notation is used. If one already understands ANOVA, this isno big deal, but for someone using the encyclopedia to learn, it is an additionaleducational burden that could have been edited out. Another example is twosections on contingency tables. Contingency Tables, in Volume 1, covers gen-eral two-dimensional tables. Two-by-Two Contingency Tables, in Volume 4,covers two-dimensional tables in which each dimension has two levels. It is not

clear why these were not presented together, and again notation is unnecessarilyinconsistent.

A last very minor criticism is the overuse of abbreviations and acronyms.At the front of each volume are six pages of abbreviations and acronyms, in-cluding everything from commonly used statistical acronyms, such as ANOVAfor analysis of variance, and OLS for ordinary least squares, to uncommonlyused acronyms (at least in statistics), such as BHP for benign prostatic hyper-trophy, BSS for blind source separation, TS for tensile strength, UCI for unob-served conditional invariance, and VLCD for very-low-calorie diet. I was oftenfrustrated by such extensive use of acronyms, and I imagine that more juniorlearners will be even more so.

Overall, I congratulate the editors for a job well done. The coverage is thor-ough and accurate, and the levels are well matched to the anticipated readers.I would encourage libraries, universities, and persons for whom a $1,800 re-source is not financially burdensome to include this encyclopedia in their col-lections.

Dalene K. STANGL

Duke University

REFERENCES

Armitage, P., and Colton, T. (eds.) (2005), Encyclopedia of Biostatistics (2nded.), Hoboken, NJ: Wiley.

Kotz, S., Read, C. B., Balakrishnan, N., and Vidakovic, B. (eds.) (2005), Ency-clopedia of Statistical Sciences (2nd ed.), Hoboken, NJ: Wiley.

Extreme Value Theory: An Introduction.

Laurens DE HAAN and Ana FERREIRA. New York: Springer, 2006. ISBN0-387-23946-4. xvi + 417 pp. $59.95.

This book is a recent addition to the Springer series on operations researchand financial engineering. It differs from most books on extreme value theoryin that it is primarily a mathematical statistics book. In contrast, for example,the fine book by Embrechts, Klüppelberg, and Mikosch (1997), develops prob-abilistic tools for modeling extremes in insurance and finance in addition toaddressing issues in statistical inference. The main objectives of de Haan andFerreira were to develop estimators for parameters of extreme value distribu-tions, study their asymptotic properties, and test their performance.

The origins of extreme value theory date to the first half of the twentiethcentury, with the work of Fisher and Tippett (1928) and Gnedenko (1943) clas-sifying the extreme value distributions as a one-parameter family. Developingestimators for the parameter and studying its asymptotic properties is a some-what more recent phenomenon. The Hill estimator and the Picklands estimator,for example, appeared only in the mid-1970s. The authors not only introducereaders to the statistical theory of extremes, but also cover recent developments.

The book is divided into three parts: Part I, on one-dimensional observations;Part II, on multidimensional observations; and Part III, where the observationsform a stochastic process. There are also two appendixes, the second on regu-lar variation. Each part begins by giving the probabilistic underpinnings of theproblem. Then estimators are proposed, and their asymptotic properties are ex-amined. For example, the emphasis of Chapter 1 is on the characterization of theextreme value theorems and their domains of attraction. The regular variationresults required are all carefully proved in an appendix. Estimators are devel-oped from the intermediate order statistics, and Chapter 2 gives the necessarybackground. Chapters 3–5 introduce various estimators. Much effort is spenton proving central limit theorems for the estimators. Typically the estimatoris asymptotically biased, and the bias is found using tools from second-orderextended variation and/or applications of uniform central limit theorems. Fi-nally, the theory is applied to data. One application used throughout the book issea-level data from the Netherlands.

The authors assume that the reader is mathematically mature and is comfort-able with mathematical statistics. Much of the material is fairly well motivated,but at times the book reads like a research article in the Annals of Statistics.Proofs are thorough and, except in a couple of places, correct. Each chapterconcludes with a set of exercises designed to test the reader’s grasp of the the-ory. Few if any of the exercises ask the reader to work with a data set. Evenwithin the subject of extreme value theory, the book is rather specialized and,except in unique situations, not an ideal textbook choice. On the other hand,for someone wanting to work in statistical aspects of extreme value theory, the

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Book Reviews 883

book is an excellent choice for self-study. The reader not only will receive a rig-orous introduction, but also will get to the cutting edge of inference for extremevalues.

Peter C. KIESSLER

Clemson University

REFERENCES

Embrechts, P., Klüppelberg, C., and Mikosch, T. (1997), Modeling ExtremalEvents for Insurance and Finance, New York: Springer.

Fisher, R. A., and Tippett, L. H. C. (1928), “Limiting Forms of the FrequencyDistribution of the Largest and Smallest Member of a Sample,” Proceedingsof the Cambridge Philosophical Society, 24, 180–190.

Gnedenko, B. V. (1943), “Sur la Distribution Limite du Terme Maximum d’uneSérie Aléatorie,” The Annals of Mathematics, 44, 423–453.

An Introduction to Bayesian Analysis: Theory and Methods.

Jayanta K. GHOSH, Mohan DELAMPADY, and Tapas SAMANTA. NewYork: Springer, 2006. ISBN 0-387-40084-2. xiii + 352 pp. $79.95.

Bayesian analysis has arrived. There are signs everywhere to suggest thatthe Bayesian paradigm is not in fact the “dark side,” but that Bayesian solutionsto important problems may offer advantages over the classical or frequentistalternatives. Perhaps the most obvious indicator of this is that even the mostclassically inclined graduate (and even some undergraduate) programs includeone or more Bayesian courses for their students. In this time of expansion, it isimperative that high-quality texts for educating students about Bayesian analy-sis be written. This text offers one approach based on the pedagogic decisionto “balance theory, methods, and applications.” In this review, I assess the ex-tent to which the authors have met their goal by briefly describing the book andindicating topics that would benefit from increased or modified coverage.

The brief introduction to classical inference presented in the first chapterprovides a nice basis for the objective Bayesian treatment offered by the au-thors throughout the book. The authors present a decision-theoretic approachto inference that lends itself to the Bayesian methods investigated in the laterchapters of the book. In some sense, the chapter covering classical inference hasa “know-thy-enemy” feel. This expository chapter presents a brief tour of thebasic concepts in classical statistics, including exponential and location-scalefamilies of distributions, maximum likelihood inference, sufficiency, point es-timation, interval estimation, and hypothesis testing. The authors have chosenan intriguing subset of the important concepts of classical statistical inference.Unfortunately, the coverage of these various aspects is uneven (five pages ofpoint estimation, five pages of hypothesis testing, and one page of interval esti-mation). I would think that from a Bayesian or classical perspective, the balancewould be shifted in nearly the opposite direction. Finally, as a Bayesian, I wasexcited to see the “Changing Face of Classical Inference” section, only to bedisappointed by the one-paragraph reference to bootstrapping.

The second chapter provides an excellent treatment of the basics of Bayesianinference, closely mirroring the presentation in the first chapter. The topics ofpoint estimation, interval estimation, and hypotheses are all presented from aBayesian, decision-theoretic point of view. The discussion is based primarilyon an objective Bayesian formulation. Several examples from classical infer-ence are presented, most from an abstract perspective with traditional ball-in-urn and coin-tossing experiments. The authors give an enlightening, albeitabridged, discussion of the advantages of being Bayesian. Their list, althoughincomplete, provides a nice motivation for the rest of the textbook.

The next three chapters focus on fairly theoretical treatments of decision the-ory (including posterior robustness), large-sample theory, and (mostly) objec-tive prior construction. The authors give an impressive presentation of this ma-terial in more detail than what I typically provide in my introductory Bayesiancourse. The coverage is reasonably complete and accurate. As the Introductionindicates, this book could be considered a candidate for a one- or two-semester(it would be difficult to cover the material presented in this textbook in onesemester) course in Bayesian analysis. The breadth of coverage is sufficient,but I would have liked to see inclusion of more intuitive explanations and mo-tivation.

Chapter 6 is concerned primarily with hypothesis testing. The chapter titlesuggests that model selection also is discussed. Although model selection is

a natural operation in a fully Bayesian treatment, hypothesis testing is some-what foreign. I would rather have seen a whole chapter on model selectiontechniques, including Bayesian goodness-of-fit techniques, and one small sub-section on Bayesian treatments of hypothesis testing. A theme of the book isthat one may “eat the Bayesian omelet without breaking the Bayesian egg.”The objective Bayesian treatment, combined with the heavy emphasis on hy-pothesis testing, makes this book appealing for classically trained statisticians.It provides an approach that is compatible with a traditional education in classi-cal statistics. Although this is foreign to my experience with Bayesian methods,it may prove useful to instructors with anaphylaxis (extreme egg allergy).

Computational issues and techniques enter the picture in Chapter 7. The au-thors cite the work by Gelfand and Smith (1990) as opening the floodgates forBayesian computation. An excellent treatment of Markov chain Monte Carlo(MCMC) dominates the exposition here. Complemented by concise treatmentsof EM algorithms and traditional Monte Carlo methods, the MCMC materialis presented in a clear and insightful manner. The authors’ addition of Rao–Blackwellization to the standard cadre of MCMC techniques is useful and in-novative. The treatment of reversible-jump MCMC is insufficient for practicalusage. I suspect that this section was included for completeness with the hopethat the reader will dig more deeply into the literature to learn about implemen-tation in the context of real problems.

The traditional topics in introductory statistics are covered from an objec-tive Bayesian approach in Chapter 8. These include comparison of two means,(multiple) linear regression, and logistic regression, as well as some variants.Although the breadth of the treatment here is appropriate, many instructors mayfind favorite topics missing, including residuals, influence measures, and analy-sis of variance (ANOVA). The authors opinion that single-factor ANOVA is asimple extension of the two-means case may be at variance with reality as per-ceived by students learning the material for the first time.

Perhaps the most compelling practical reason to be a Bayesian is the powerof hierarchical models. Under the guise of “high-dimensional problems,” hier-archical models find their place in Chapter 9. Using the canonical example fromStein (1955), the authors illustrate the important principle of shrinkage. Thespecific methods for incorporating shrinkage are empirical Bayes formulationsof various types. Hierarchical modeling is illustrated as parametric empiricalBayes. Although technically correct, disguising hierarchical modeling as onevariety of empirical Bayes seems to hide the real muscle of these models as adata-analytic tool.

The final chapter is devoted to more specialized and advanced applicationsof Bayesian analysis, including Bayesian approaches to spatial modeling, “non-parametric” regression in the form of wavelets, and Dirichlet process modeling.These brief vignettes (the entire chapter contains only 13 pages) hopefully willwhet the reader’s appetite for more detail, which can be found in referencedsources.

As someone who teaches this course on a regular basis to advanced un-dergraduates and first-year graduate students, my concerns in considering thisbook for adoption include deemphasis of posterior predictive distributions, pre-sentation of objective Bayes as the standard, and overemphasis on hypothe-sis testing. First, as a classically trained Bayesian, I find that one of the mostcompelling notions of the Bayesian paradigm is that prediction is “built-in.”A fundamental issue in scientific pursuits is accurate prediction. The authorspresent very little in the way of illustration of the power inherent in posteriorpredictive distributions. Second, although objective Bayes (OB) has an impor-tant place in Bayesian methods, my “old school”–trained mind is surprised tofind OB presented as the standard Bayesian approach. The authors acknowl-edge the importance of using subjective priors when available, but advocategoing OB primarily because subjective priors “seem difficult.” In my view, thepayoff for using subjective priors is worth the extra effort to elicit them. Finally,the authors spend a substantial amount of effort and pages exploring hypothesistesting. A personal bias is that the hypothesis testing quagmire is overempha-sized in classical inference courses, and that the Bayesian framework offers abridge, which may eliminate the need to ever traverse this precarious landscape.

Overall, I congratulate the authors for a largely successful attempt to intro-duce true religion. Their goal of balancing theory, methods, and applications iscertainly lofty, though difficult to achieve without a more expansive treatmentof the subject. Although I do not see this book supplanting the more applica-tions oriented text by Gelman, Carlin, Stern, and Rubin (2004), it would servenicely as a text for a second course in Bayesian statistics if the material wereselected judiciously.

C. Shane REESE

Brigham Young University

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