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. |
Contents
| 1 | |
Methods of Constructing Copulas | 51 |
Archimedean Copulas | 109 |
Dependence 157 | 156 |
Additional Topics | 227 |
List of Symbols | 263 |
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Common terms and phrases
analogous Archimedean copulas association assume bivariate bounds called chapter completely component concordance consequence consider construct continuous random variables convex Corollary curve Cuv uv decreasing defined Definition denote dependence derivable discussed equal equivalent Example Exercise exists expression extended F and G family of copulas Fréchet-Hoeffding function H given graph hence holds Ï Ì ifand illustrated implies increasing independent inequality integer joint distribution function Kendall’s tau Lemma limit margins measure Methods monotonicity multivariate n-copula nondecreasing Note Ô Ô obtain ordered pair parameter positive probability proof properties Prove PX xY quasi-copula region respectively sample satisfies Schweizer Sect Show shuffle similar similarly Sklar statistics strictly subset Suppose symmetric Table tail Theorem tion uniform 0,1 X and Y yields zero
Popular passages
Page 256 - JACKSON . A User's Guide to Principle Components JOHN . Statistical Methods in Engineering and Quality Assurance JOHNSON . Multivariate Statistical Simulation JOHNSON and BALAKRISHNAN . Advances in the Theory and Practice of Statistics: A Volume in Honor of Samuel Kotz JUDGE, GRIFFITHS, HILL.
Page 3 - For any * ^ 0, the value d(p, q) at x can be interpreted as "the probability that the distance between p and q is less than x"; it was approach of K.
Page 267 - Sklar, A. (1974) Operations on distribution functions not derivable from operations on random variables, Studia Math. 52, 43-52.
Page 4 - In their words, since ... under almost surely increasing transformations of (the random variables), the copula is invariant while the margins may be changed at will, it follows that it is precisely the copula which captures those properties of the joint distribution which are invariant under almost surely strictly increasing transformations.
References to this book
Comparison Methods for Stochastic Models and Risks Alfred Müller,Dietrich Stoyan No preview available - 2002 |