Extreme Value Distributions: Theory and Applications

Rating:
Summary: great overview of theory and historical development
Review: The theory of extremes and related topics of outlier detection are near and dear to me. My Ph.D. thesis dealt with special stationary sequences and how Gnedenko's three type theorem extended from the i.i.d. case to stationary sequences. Also after getting my Ph.D. I went to Oak Ridge National Laboratory where I continued to do research on extremes and focused on the problem of outlier detection and its relation to data validation.

There have been a number of theoretical books on the theory of extremes. Gumbel's was the first and it was also considered very practical. Galambos and Leadbetter,Lindgren and Rootzen are the landmark theoretical books that extend the theory. Castillo wrote a book for engineers and in recent years there have been additional books of a theoretical nature but with applications to finance or hydrology etc.

This book is a reference for engineers and others that may want to use the distributions and apply the methods. It is also useful to the extreme value theorists like myself.

Kotz has always been known for his ability to characterize and provide historical accounts for probability distributions. His volumes with Norman Johnson are famous. This book is up to his high standards. Chapter 1 provides a thorough and accurate historical account of the development of the theory of univariate extremes. Chapter 2 provides information on the family of generalized extreme value distributions. Finally Chapter 3 covers the theory and distributions for multivariate extremes along with applications. This is followed by a large and extensive bibliography. Although my own work is not mentioned directly in this book it can be found in some of the references including Castillo (1988) and Leadbetter, Lindgren and Rootzen (1983).

Rating:
Summary: great overview of theory and historical development
Review: The theory of extremes and related topics of outlier detection are near and dear to me. My Ph.D. thesis dealt with special stationary sequences and how Gnedenko's three type theorem extended from the i.i.d. case to stationary sequences. Also after getting my Ph.D. I went to Oak Ridge National Laboratory where I continued to do research on extremes and focused on the problem of outlier detection and its relation to data validation.

There have been a number of theoretical books on the theory of extremes. Gumbel's was the first and it was also considered very practical. Galambos and Leadbetter,Lindgren and Rootzen are the landmark theoretical books that extend the theory. Castillo wrote a book for engineers and in recent years there have been additional books of a theoretical nature but with applications to finance or hydrology etc.

This book is a reference for engineers and others that may want to use the distributions and apply the methods. It is also useful to the extreme value theorists like myself.

Kotz has always been known for his ability to characterize and provide historical accounts for probability distributions. His volumes with Norman Johnson are famous. This book is up to his high standards. Chapter 1 provides a thorough and accurate historical account of the development of the theory of univariate extremes. Chapter 2 provides information on the family of generalized extreme value distributions. Finally Chapter 3 covers the theory and distributions for multivariate extremes along with applications. This is followed by a large and extensive bibliography. Although my own work is not mentioned directly in this book it can be found in some of the references including Castillo (1988) and Leadbetter, Lindgren and Rootzen (1983).

Rating:
Summary: Too many errors and omissions
Review: This book is a broad survey of both univariate and multivariate extreme value distributions. The target reader is someone who wants quick access to results so the book has no proofs, although there are motivating theoretical discussions in the first chapter. This book has considerable overlap with a chapter in Kotz's book: Continuous Univariate Distributions, Vol. 2. The book is thin (too thin for such a vast subject) with only three chapters. There are no worked problems or sample data sets, so the person wanting a tutorial work should look elsewhere. This book is primarily a reference work, not a text, or a monograph. Unfortunately, there are a number of annoying errors and omissions. For example, on page 23 (section 1.7.1 Moment Estimation) the moment equations for sigma and mu reversed. In chapter 2, which covers generalized extreme value distributions, the authors reference Castillo and Hadi (1997), but this reference is missing from the bibliography. Too bad, as this is an essential reference for the chapter because as it gives a good method for estimating the parameters of a generalized extreme value distribution. Although there is a section on Bayesian inference, try looking up "Bayes" or "Bayesian" in the index-- nothing. The index is entirely inadequate, especially for a reference work. If I had paid for this book with my own money, I would have returned it.

The statistics of extremes has a variety of applications including: civil engineering, reliability, meteorology, seismology, hydrology, insurance, and finance, to name but a few. The classic reference has been the book by Gumbel-- a tour-de-force of the subject. However, that book predates modern developments in numerical methods so it is dated. Castillo (1988) updates Gumbel with a number of results that don't even appear in the Kotz book.

I wish I could recommend "Extreme Value Distributions" because of the many fine other books authored by Kotz, but I can't. Unfortunately, it looks like it was put together in a hurry, the reader deserves better.