You are here

Laws of Small Numbers: Extremes and Rare Events

Michael Falk, Jürg Hüsler, and Rolf-Dieter Reiss
Publication Date: 
Number of Pages: 
[Reviewed by
Gudmund Iversen
, on

George Bernhard Shaw is thought to have said: "Everything happens to everybody sooner and later if there is time enough." He clearly would have understood this book.

The law of small numbers refers to the Poisson approximation of the binomial distribu-tion with a small probability of an event occurring. Von Bortkiewicz first named the law in 1898, and his example on deaths of cavalrymen by horse-kicks in the Prussian army is known to legions of students.

The book consists of three parts. The first part deals with a generalization of the Poisson process as well as an introduction to extreme value theory, focusing on the independent and identically distributed case. The second part takes up the study of multivariate ex-tremes, and the third part deals with non-identically and independent distributed variables.

While the book deals with a topic of interest to many students of probability, it may still end up having a small distribution. It is heavily mathematical, dealing with a specialized topic, and while there are many examples, there are no exercises. That would limit the use of the book as a textbook. Still, it is valuable to have this material gathered in one place.

Gudmund R. Iversen received his PhD in statistics from Harvard University and taught statistics for many years at The University of Michigan and Swarthmore College until his retirement.

 Functional Laws of Small Numbers.- ExtremeValueTheory.- Estimation of Conditional Curves.- Basic Theory of Multivariate Maxima.- Multivariate Extremes: The Pickands Approach.- The Pickands Approach in the Bivariate Case.- Multivariate Extremes: Supplementary Concepts and Results.- Introduction to the Non IID Case.- Extremes of Random Sequences.- Extremes of Gaussian Processes.- Extensions for Rare Events.- Statistics of Extremes.