Statistics of Extremes

E. J. Gumbel

• Publisher: Dover Publications
• Year: 2012
• Print ISBN: 9780486436043
• eBook ISBN: 9780486154480
• Topic: Mathematics
• Subtopic: Probability & Statistics
• Index: Mathematics

Universally acknowledged as the classic text about statistics of extremes, this volume is geared toward use by statisticians and statistically minded scientists and engineers. It employs elementary terms to explain applications, favors graphical procedures over calculations, and presents simple generalizations as exercises — all of which contribute to its value for students. Starting with definitions of its aims and tools, the text proceeds to discussions of order statistics and their exceedances, exact distribution of extremes, and analytical study of extremes. Additional topics include the first asymptotic distribution; uses of the first, second, and third asymptotes; and the range. 1958 edition. 44 tables. 97 graphs.

Chapter One: AIMS AND TOOLS

“Blessed is he who expects nothing for he shall never be disappointed.”

1.0. AIMS

1.0.1. Conditions.

The aim of a statistical theory of extreme values is to analyze observed extremes and to forecast further extremes. The extremes are not fixed, but are new statistical variates depending upon the initial distribution and upon the sample size. However, certain results which are distribution-free may be reached.

Statistical studies of extreme values are meant to give an answer to two types of questions: 1) Does an individual observation in a sample taken from a distribution, alleged to be known, fall outside what may reasonably be expected? 2) Does a series of extreme values exhibit a regular behavior? In both cases, “reasonable expectation” and “regular behavior” have to be defined by some operational procedure.

The essential condition in the analysis is the “clausula rebus sic stantibus.” The distribution from which the extremes have been drawn and its parameters must remain constant in time (or space), or the influence that time (or space) exercises upon them must be taken into account or eliminated. Another limitation of the theory is the condition that the observations from which the extremes are taken should be independent. This assumption, made in most statistical work, is hardly ever realized. However, if the conditions at each trial are determined by the outcome of previous trials as, e.g., in Pólya’s and Markoff’s schemes, some classical distributions are reached. Therefore, the assumption that distributions. based on dependent events should share the asymptotic properties of distributions based on independent trials is not too far-fetched.

One of the principal notions to be used is the “unlimited variate.” Here “common sense” revolts at once, and practical people will say: “Statistical variates should conform to physical realities, and infinity transcends reality. Therefore this assumption does not make sense.” The author has met this objection when he advocated his theory of floods (6.3.1), at which time this issue was raised by people who applied other unlimited distributions without realizing that their methods rely on exactly the same notion.

This objection is not as serious as it looks, since the denial of the existence of an upper or lower limit is linked to the affirmation that the probability for extreme values differs from unity (or from zero) by an amount which becomes as small as we wish. Distributions currently used have this property. The exploration of how unlimited distributions behave at infinity is just part of the common general effort of mathematics and science to transgress the finite, as calculus has done since Newton’s time for the infinite, and nuclear physics is doing for the infinitesimal.

References: Borel, Chandler.

1.0.2. History.

The founders of the calculus of probabilities were too occupied with the general behavior of statistical masses to be interested in the extremes. However,¹ as early as 1709 Nicolaus Bernoulli considers an actuarial problem: n men of equal age die within t years. What is the mean duration of life of the last survivor? He reduces this question to the following: n points lie at random on a straight line of length t. Then he calculates the mean largest distance from the origin.

The first researches pertaining to the theory of largest values started from the normal distribution. This was reasonable in view of its ...

Table of contents

• Title Page
• Copyright Page
• Dedication
• PREFACE
• Table of Contents
• List of Tables
• Chapter One: AIMS AND TOOLS
• Chapter Two: ORDER STATISTICS AND THEIR EXCEEDANCES
• Chapter Three: EXACT DISTRIBUTION OF EXTREMES
• Chapter Four: ANALYTICAL STUDY OF EXTREMES
• Chapter Five: THE FIRST ASYMPTOTIC DISTRIBUTION
• Chapter Six: USES OF THE FIRST ASYMPTOTE
• Chapter Seven: THE SECOND AND THIRD ASYMPTOTES
• Chapter Eight: THE RANGE
• SUMMARY
• BIBLIOGRAPHY
• INDEX