## An Introduction to CopulasCopulas 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. |

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### Contents

Introduction | 1 |

Definitions and Basic Properties | 5 |

22 Copulas | 8 |

23 Sklars Theorem | 14 |

24 Copulas and Random Variables | 21 |

25 The FréchetHoeffding Bounds for Joint Distribution Functions of Random Variables | 26 |

26 Survival Copulas | 28 |

27 Symmetry | 31 |

44 Order and Limiting Cases | 108 |

45 Twoparameter Families | 114 |

46 Multivariate Archimedean Copulas | 121 |

Dependence | 125 |

52 Dependence Properties | 151 |

53 Other Measures of Association | 169 |

54 Median Regression | 175 |

55 Empirical Copulas | 176 |

28 Order | 34 |

29 Random Variate Generation | 35 |

210 Multivariate Copulas | 37 |

Methods of Constructing Copulas | 45 |

32 Geometric Methods | 52 |

33 Algebraic Methods | 79 |

34 Constructing Multivariate Copulas | 84 |

Archimedean Copulas | 89 |

42 Oneparameter Families | 93 |

56 Multivariate Dependence | 179 |

Additional Topics | 183 |

62 Operations on Distribution Functions | 191 |

63 Markov Processes | 194 |

201 | |

List of Symbols | 211 |

213 | |

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### Common terms and phrases

2-increasing absolutely continuous Ali-Mikhail-Haq family analogous Archimedean copulas bivariate distribution bution function C-measure Chapman-Kolmogorov equations chimedean completely monotonic component construct continuous random variables Corollary cubic sections defined Definition denote dependence properties diagonal section equivalent Example Exercise exponential distribution F and G families of Archimedean family of copulas following theorem Frechet function H Genest given graph Hence inequality joint distribution function Kendall's tau Lemma Let H level curve likelihood ratio dependence line segments marginal distribution margins F Markov process measure of concordance measure of dependence measures of association multivariate n-copula n-dimensional Nelsen nondecreasing nonincreasing Note ordinal sum pair parameter population version radial symmetry rectangle respectively satisfies Show shuffle Sklar's theorem Spearman's rho statistics stochastic strictly increasing subcopula subset survival copula survival function symmetric tail monotonicity tion uniform 0,1 univariate variables whose copula variables whose joint variables with copula variables with joint yields

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### References to this book

Comparison Methods for Stochastic Models and Risks Alfred Müller,Dietrich Stoyan No preview available - 2002 |