An Introduction to Copulas

226 Pages·1999·6.861 MB·English

by  Roger B. Nelsen (auth.)

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Lecture Notes in Statistics 139
Edited by P. Bickel, P. Diggle, S. Fienberg, K. Krickeberg,
I. OIkin, N. Wennuth, S. Zeger
Springer Science+Business Media, LLC
Roger B. Nelsen
An Introduction to Copulas
~ Springer
Roger B. Nelsen
Department of Mathematical Sciences
Lewis & Clark College
Portland, OR 97219-7899
USA
Library of Congress Cataloging-In-Publication Data
Nelsen, Roger B.
An Introduction to copulas I Roger B. Nelsen.
p. c •. -- (Lecture notes In statistics; 139)
Includes bibliographical references and Index.
ISBN 978-0-387-98623-4 ISBN 978-1-4757-3076-0 (eBook)
DOI 10.1007/978-1-4757-3076-0
1. Copulas (Mathe.atlcal statistics) I. Title. II. Series:
Lecture notes In statistics (Springer-Verlag) ; v. 139.
CA273.6.N45 1998
519.5'35--dc21 98-41245
ISBN 978-0-387-98623-4 Printed on acid-free paper
© 1999 Springer Science+Business Media New York
Originally published by Springer-Verlag New York, Inc. in 1999
Softcover reprint of the hardcover 1st edition 1999
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To my mother and father
Preface
In November of 1995, I was at the University of Massachusetts in Amherst for a
few days to attend a symposium held, in part, to celebrate Professor Berthold
Schweizer's retirement from classroom teaching. During one afternoon break, a
small group of us were having coffee following several talks in which copulas
were mentioned. Someone asked what one should read to learn the basics about
copulas. We mentioned several references, mostly research papers and conference
proceedings. I then suggested that perhaps the time was ripe for "someone" to
write an introductory-level monograph on the subject. A colleague, I forget who,
responded somewhat mischievously, "Good idea, Roger-why don't you write
it?"
Although flattered by the suggestion, I let it lie until the following Septem
ber, when I was in Prague to attend an international conference on distributions
with fixed marginals and moment problems. In Prague, I asked Giorgio
Dall' Aglio, Ingram Olkin, and Abe Sklar if they thought that there might indeed
be interest in the statistical community for such a book. Encouraged by their
responses and knowing that I would soon be eligible for a sabbatical, I began to
give serious thought to writing an introduction to copulas.
This book is intended for students and practitioners in statistics and probabil
ity-at almost any level. The only prerequisite is a good upper-level undergradu
ate course in probability and mathematical statistics, although some background
in nonparametric statistics would be beneficial. Knowledge of measure-theoretic
probability is not required.
The book begins with the basic properties of. copulas, and then proceeds to
present methods for constructing copulas, and to discuss the role played by copu
las in modeling and in the study of dependence. The focus is on bivariate copu
las, although most chapters conclude with a discussion of the multivariate case.
As an introduction to copulas, it is not an encyclopedic reference, and thus it is
necessarily incomplete-many topics which could have been included are omit
ted. The reader seeking additional material on families of continuous bivariate
distributions and their applications should see [Hutchinson and Lai (1990)]; arxl
the reader interested in learning more about multivariate copulas and dependence
should consult [Joe (1997)].
There are about 150 exercises in the book. While it is certainly not necessary
to do all (or indeed any) of them, the reader is encouraged to read through the
statements of the exercises before proceeding to the next section or chapter. Al
though some exercises do not add anything to the exposition (e.g., "Prove Theo-
viii An Introduction to Copulas
rem 1.1.1 "), many present examples, counterexamples, and supplementary topics
which are often referenced in subsequent sections.
I would like to thank Lewis & Clark College for granting me a sabbatical
leave in order to write this book; and my colleagues in the Department of
Mathematics, Statistics, and Computer Science at Mount Holyoke College for
graciously inviting me to spend the sabbatical year with them. Thanks too to
Ingram Olkin for suggesting and encouraging that I consider publication with
Springer's Lecture Notes in Statistics; and to John Kimmel, the executive editor
for statistics at Springer, for his valuable assistance in the publication of this
book.
Finally, I would like to express by gratitude and appreciation to all those
with whom I have had the pleasure of working on problems related to copulas
and their applications: Claudi Alsina, Jerry Frank, Greg Fredricks, Juan Quesada
Molina, Jose Antonio Rodriguez Lallena, Carlo Sempi, and Abe Sklar. But
most of all I want to thank my good friend and mentor Berthold Schweizer, who
not only introduced me to the subject, but who has consistently and unselfishly
aided me in the years since, and who inspired me to write this book. I also want
to thank Bert for his careful and critical reading of earlier drafts of the manu
script, and his invaluable advice on matters mathematical and stylistic. However,
it goes without saying that any and all remaining errors in the book are mine
alone.
Roger B. Nelsen
Portland, Oregon
July 1998
Contents
Preface vii
1 Introduction 1
2 Definitions and Basic Properties 5
2.1 Preliminaries. . . 5
2.2 Copulas. . . . . 8
Exercises 2.1-2.11 12
2.3 Sklar's Theorem . 14
2.4 Copulas and Random Variables. 21
Exercises 2.12-2.17. . . . . . 24
2.5 The Frechet-Hoeffding Bounds for Joint Distribution Functions
of Random Variables 26
2.6 Survival Copulas .. 28
Exercises 2.18-2.25 . 30
2.7 Symmetry. . . . . 31
2.8 0n:Ier ...... . 34
Exercises 2.26-2.32 . 34
2.9 Random Variate Generation 35
2.10 Multivariate Copulas 37
Exercises 2.33-2.36. . . 43
3 Methods of Constructing Copulas 45
3.1 The Inversion Method. . 45
3.1.1 The Marshall-Olkin Bivariate Exponential Distribution 46
3.1.2 The Circular Uniform Distribution 48
Exercises 3.1-3.6. . . . . . . . . . . . . . . 50
3.2 Geometric Methods. . . . . . . . . . . . . . 52
3.2.1 Singular Copulas with Prescribed Support. 52
3.2.2 Ordinal Sums . . . . . . . . . . . . . 55
x An Introduction to Copulas
Exercises 3.7-3.13 . 56
3.2.3 Shuffles of M 59
3.2.4 Convex Sums 64
Exercises 3.14-3.20. 66
3.2.5 Copulas with Prescribed Horizontal or Vertical Sections. 67
3.2.6 Copulas with Prescribed Diagonal Sections 74
Exercises 3.21-3.33. . . . . 76
3.3. Algebraic Methods . . . . . . . . . 79
3.3.1 Plackett Distributions. . . . . 79
3.3.2 Ali-Mikhail-Hag Distributions. 81
Exercises 3.34-3.41. . . . . . . . 83
3.4 Constructing Multivariate Copulas. 84
4 Archimedean Copulas 89
4.1 Definitions . . . . 89
4.2 One-parameter Families . 93
4.3 Fundamental Properties . 93
Exercises 4.1-4.15 . . . 106
4.4 Order and Limiting Cases 108
4.5 Two-parameter Families. 114
4.5.1 Families of Generators 114
4.5.2 Rational Archimedean Copulas. 117
Exercises 4.16-4.21. . . . . . . 120
4.6 Multivariate Archimedean Copulas 121
5 Dependence 125
5. 1 Concordance. 125
5.1.1 Kendall's tau. 126
Exercises 5.1-5.5. . 133
5.1.2 Spearman's rho. 134
Exercises 5.6-5.14 .. 138
5.1.3 The Relationship between Kendall's tau
and Spearman's rho ..... 141
5.1.4 Other Concordance Measures. 146
Exercises 5.15-5.20. . . . 149
5.2 Dependence Properties. . . 151
5.2.1 Quadrant Dependence 151
Exercises 5.21-5.27 .... 153