An Introduction to Copulas,9780387986234

An Introduction to Copulas

by
ISBN13: 9780387986234
ISBN10: 0387986235
Format: Paperback
Pub. Date: 1999-06-01
Publisher(s): Springer Verlag
List Price: $83.95

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Summary

Copulas are functions that join multivariate distribution functions to their one-dimensional margins. The study of copulas and their role in statistics is a new but vigorously growing field. In this book the student or practitioner of statistics and probability will find discussions of the fundamental properties of copulas and some of their primary applications. The applications include the study of dependence and measures of association, and the construction of families of bivariate distributions.With nearly a hundred examples and over 150 exercises, this book is suitable as a text or for self-study. The only prerequisite is an upper level undergraduate course in probability and mathematical statistics, although some familiarity with nonparametric statistics would be useful. Knowledge of measure-theoretic probability is not required.Roger B. Nelsen is Professor of Mathematics at Lewis & Clark College in Portland, Oregon. He is also the author of "Proofs Without Words: Exercises in Visual Thinking," published by the Mathematical Association of America.

Table of Contents

Preface vii
1 Introduction
1(4)
2 Definitions and Basic Properties
5(40)
2.1 Preliminaries
5(3)
2.2 Copulas
8(6)
Exercises 2.1-2.11
12(2)
2.3 Sklar's Theorem
14(7)
2.4 Copulas and Random Variables
21(5)
Exercises 2.12-2.17
24(2)
2.5 The Frechet-Hoeffding Bounds for Joint Distribution Functions of Random Variables
26(2)
2.6 Survival Copulas
28(3)
Exercises 2.18-2.25
30(1)
2.7 Symmetry
31(3)
2.8 Order
34(1)
Exercises 2.26-2.32
34(1)
2.9 Random Variate Generation
35(2)
2.10 Multivariate Copulas
37(8)
Exercises 2.33-2.36
43
3 Methods of Constructing Copulas
45(44)
3.1 The Inversion Method
45(7)
3.1.1 The Marshall-Olkin Bivariate Exponential Distribution
46(2)
3.1.2 The Circular Uniform Distribution
48(2)
Exercises 3.1-3.6
50(2)
3.2 Geometric Methods
52(27)
3.2.1 Singular Copulas with Prescribed Support
52(3)
3.2.2 Ordinal Sums
55(1)
Exercises 3.7-3.13
56(3)
3.2.3 Shuffles of M
59(5)
3.2.4 Convex Sums
64(2)
Exercises 3.14-3.20
66(1)
3.2.5 Copulas with Prescribed Horizontal or Vertical Sections
67(7)
3.2.6 Copulas with Prescribed Diagonal Sections
74(2)
Exercises 3.21-3.33
76(3)
3.3. Algebraic Methods
79(5)
3.3.1 Plackett Distributions
79(2)
3.3.2 Ali-Mikhail-Haq Distributions
81(2)
Exercises 3.34-3.41
83(1)
3.4 Constructing Multivariate Copulas
84(5)
4 Archimedean Copulas
89(36)
4.1 Definitions
89(4)
4.2 One-parameter Families
93(1)
4.3 Fundamental Properties
93(15)
Exercises 4.1-4.15
106(2)
4.4 Order and Limiting Cases
108(6)
4.5 Two-parameter Families
114(7)
4.5.1 Families of Generators
114(3)
4.5.2 Rational Archimedean Copulas
117(3)
Exercises 4.16-4.21
120(1)
4.6 Multivariate Archimedean Copulas
121(4)
5 Dependence
125(58)
5.1 Concordance
125(26)
5.1.1 Kendall's tau
126(7)
Exercises 5.1-5.5
133(1)
5.1.2 Spearman's rho
134(4)
Exercises 5.6-5.14
138(3)
5.1.3 The Relationship between Kendall's tau and Spearman's rho
141(5)
5.1.4 Other Concordance Measures
146(3)
Exercises 5.15-5.20
149(2)
5.2 Dependence Properties
151(18)
5.2.1 Quadrant Dependence
151(2)
Exercises 5.21-5.27
153(2)
5.2.2 Tail Monotonicity
155(4)
5.2.3 Stochastic Monotonicity, Corner Set Monotonicity and Likelihood Ratio Dependence
159(8)
Exercises 5.28-5.37
167(2)
5.3 Other Measures of Association
169(6)
5.3.1 Measures of Dependence
169(4)
5.3.2 Measures Based on Gini's Coefficient
173(1)
Exercises 5.38-5.44
174(1)
5.4 Median Regression
175(1)
5.5 Empirical Copulas
176(3)
5.6 Multivariate Dependence
179(4)
6 Additional Topics
183(18)
6.1 Distributions with Fixed Margins
183(8)
Exercises 6.1-6.5
189(2)
6.2 Operations on Distribution Functions
191(3)
6.3 Markov Processes
194(7)
Exercises 6.6-6.10
198(3)
References 201(10)
List of Symbols
211(2)
Index 213

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