INSEAD Library

Includes bibliographical references and index

Digitized

An Introduction to Copulas Contents Preface to the First Edition.......................................................................................... vii Preface to the Second Edition...................................................................................... ix 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