1 Introduction 1
2 Definitions and Basic Properties 7
2.1 Preliminaries 7
2.2 Copulas 10
Exercises 2.1-2.11 14
2.3 Sklar’s Theorem 17
2.4 Copulas and Random Variables 24
Exercises 2.12-2.17 28
2.5 The Fréchet-Hoeffding Bounds for Joint Distribution
Functions of Random Variables 30
2.6 Survival Copulas 32
Exercises 2.18-2.26 34
2.7 Symmetry 36
2.8 Order 38
Exercises 2.27-2.33 39
2.9 Random Variate Generation 40
2.10 Multivariate Copulas 42
Exercises 2.34-2.37 48
3 Methods of Constructing Copulas 51
3.1 The Inversion Method 51
3.1.1 The Marshall-Olkin Bivariate Exponential Distribution 52
3.1.2 The Circular Uniform Distribution 55
Exercises 3.1-3.6 57
3.2 Geometric Methods 59
3.2.1 Singular Copulas with Prescribed Support 59
3.2.2 Ordinal Sums 63
Exercises 3.7-3.13 64
3.2.3 Shuffles of M 67
3.2.4 Convex Sums 72
Exercises 3.14-3.20 74
3.2.5 Copulas with Prescribed Horizontal or Vertical Sections 76
3.2.6 Copulas with Prescribed Diagonal Sections 84
xii Contents
Exercises 3.21-3.34 86
3.3 Algebraic Methods 89
3.3.1 Plackett Distributions 89
3.3.2 Ali-Mikhail-Haq Distributions 92
3.3.3 A Copula Transformation Method 94
3.3.4 Extreme Value Copulas 97
Exercises 3.35-3.42 99
3.4 Copulas with Specified Properties 101
3.4.1 Harmonic Copulas 101
3.4.2 Homogeneous Copulas 101
3.4.3 Concave and Convex Copulas 102
3.5 Constructing Multivariate Copulas 105
4 Archimedean Copulas 109
4.1 Definitions 109
4.2 One-parameter Families 114
4.3 Fundamental Properties 115
Exercises 4.1-4.17 132
4.4 Order and Limiting Cases 135
4.5 Two-parameter Families 141
4.5.1 Families of Generators 141
4.5.2 Rational Archimedean Copulas 146
Exercises 4.18-4.23 150
4.6 Multivariate Archimedean Copulas 151
Exercises 4.24-4.25 155
5 Dependence 157
5.1 Concordance 157
5.1.1 Kendall’s tau 158
Exercises 5.1-5.5 165
5.1.2 Spearman’s rho 167
Exercises 5.6-5.15 171
5.1.3 The Relationship between Kendall’s tau
and Spearman’s rho 174
5.1.4 Other Concordance Measures 180
Exercises 5.16-5.21 185
5.2 Dependence Properties 186
5.2.1 Quadrant Dependence 187
Exercises 5.22-5.29 189
5.2.2 Tail Monotonicity 191
5.2.3 Stochastic Monotonicity, Corner Set Monotonicity,
and Likelihood Ratio Dependence 195
Exercises 5.30-5.39 204
5.3 Other Measures of Association 207
5.3.1 Measures of Dependence 207
5.3.2 Measures Based on Gini’s Coefficient 211
Exercises 5.40-5.46 213
5.4 Tail Dependence 214
Contents xiii
Exercises 5.47-5.50 216
5.5 Median Regression 217
5.6 Empirical Copulas 219
5.7 Multivariate Dependence 222
6 Additional Topics 227
6.1 Distributions with Fixed Margins 227
Exercises 6.1-6.5 233
6.2 Quasi-copulas 236
Exercises 6.6-6.8 240
6.3 Operations on Distribution Functions 241
6.4 Markov Processes 244
Exercises 6.9-6.13 248
References 251
List of Symbols 263
Index 265