Elements of Copula Modeling with RThis book introd

文件名称: Elements of Copula Modeling with R

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详细说明：This book introduces the main theoretical findings related to copulas and shows how statistical modeling of multivariate continuous distributions using copulas can be carried out in the R statistical environment with the package copula (among others). Copulas are multivariate distribution functions with standard uniform univariate margins. They are increasingly applied to modeling dependence among random variables in fields such as risk management, actuarial science, insurance, finance, engineering, hydrology, climatology, and meteorology, to name a few.Moreinformationaboutthisseriesathttp://www.springer.com/series/6991 Marius hofer Ivan Kojadinovic Martin macher. Jun Yan Elements of Copula Modeling with r ② Springer Marius hofer Ivan Kojadinovic Department of Statistics and actuarial Laboratory of Mathematics and its Science Applications University of waterloo University of pau and pays de l'adour Waterloo. Ontario Canada Pau. france Martin macher un ran Seminar for statistics Department of Statistic ETH Zurich University of Connecticut Zurich Switzerland Storrs. Connecticut. USA ISSN2197-5736 issn 2197-5744(electronic) Use r ISBN978-3-319-89634-2 ISBN978-3-319-89635-9( e Book) https://doi.org/10.10071978-3-319-89635-9 Library of Congress Control Number: 2018940269 Mathematics Subject Classification(2010): 62H05, 65C10, 62H12, 62H15, 62P05, 62P12, 65C60 O Springer International Publishing AG, part of Springer Nature 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse ll,6330 Cham, Switzerland To saisai Nire emazteari eta haurrei, maitasunez To Liselotte, with thanks for many hours of waitin lg To Jiafeng Bohan, and Jolin, with love Preface The aim of this book is to show how some of the main steps involved in the statistical modeling of continuous multivariate distributions using copulas can be carried out in the R statistical environment(R Core Team, 2017)with the R package copula The R package copula originally emerged from the authors'research interests it has been available on the Comprehensive r Archive Network(CRAN)since 2005 and has been under constant development ever since. Judging from users feedback, the package is applied in areas such as hydrology, environmental sciences, quantitative risk management, insurance, and finance Compared to existing monographs on copulas, the originality of this book is to illustrate how theoretical concepts can be applied in practice with the R package copula. To this end, numerous stand-alone examples are provided. This is both of pedagogical and of practical interest as the R source code and reproduced outputs such as figures not only allow one to better understand the subtleties of the theoretical notions but also enable one to solve real-world problems in the various fields of interest The book targets(possibly future) statisticians and(financial, hydrological, and other) engineers alike who would like to understand how theoretical notions and practical computations around copula modeling can be applied in R without an overwhelming amount of mathematics Readers are nonetheless expected to have a basic knowledge of(multivariate) probability and statistics, in particular of random vectors, simulation algorithms, estimation methods, and statistical tests Although the book addresses most of the practical issues arising when modeling multivariate data with copulas, it only covers a modest part of the field. In particular, important aspects such as dynamic copula models, vine copulas, and copula modeling for censored or discontinuous data are not dealt with. This is partly due to a selection bias: Concepts that are not implemented in the r package copula at the time of writing are not presented in the book. Many of these more specialized notions are, however, available in other R packages. For an up-to-date description of the functionalities of other packages dealing with copulas, see, for example thecranViewdistributions'aVailableathttps:/cran.r-project.org/web/views/ Distributions. html VIl Preface As the r statistical environment and most free software the r package copula comes with no warranty It is distributed with the hope that it can be useful to others Waterloo oN. Canada Marius hofer Pau. france Ivan Kojadinovic Zurich Switzerland Martin mahler Storrs CT USA Jun Yan November 2017 Reference R Core Team (2017). R: A Language and Environment for Statistical Computing. R Foundation forStatisticalComputing,ViennaAustria.https://www.r-project.org Contents 1 Introduction 1.1 A Motivating Example.......... 1.2 Probability and Quantile Transformations 1.3 Copulas 1. 4 Structure and philosophy of the book 1. 5 Additional reading........................... References 113567899 2 Copulas 2.1 Definition and Characterization 2.2 The Frechet-Hoeffding Bounds 18 2.3 Sklar’ s Theoren. 2.4 The Invariance Principle..…………,35 2.5 Survival Copulas and Copula symmetries 40 2.6 Measures of association 2.6.1 Fallacies related to the Correlation Coefficient 46 2.6.2 Rank Correlation measures 51 2.6.3 Tail Dependence Coefficients................ 58 2.7 Rosenblatt Transform and Conditional Sampling........... 68 References 3 Classes and Families 81 3.1 Elliptical Distributions and Copulas 81 3.1.1 Elliptical distributions 3.1.2 Elliptical copulas 85 3.2 Archimedean Copulas 3.3 Extreme-Value Copulas 112 34 Selected Copula Transformations and Constructions……….1l7 3.4.1 Rotated copulas 117 3.4.2 Khoudraji's device 120 34.3 Mixtures of Copulas∴…… 127 References 130 Contents 4 Estimation 133 4. 1 Estimation Under a Parametric Assumption on the Copula 133 4.1.1 Parametrically Estimated Margins 134 4.1.2 Nonparametrically Estimated Margins 4.1.3 Estimators of Elliptical Copula Parameters..........148 4.1.4 Other Semi-parametric Estimators ............... 152 4.1.5 Estimation of Copula Models with Partly Fixed Parameters 153 4.2 Nonparametric Estimation of the Copula 157 4.2.1 The Empirical Copula 158 4.2.2 Under Extreme-Value Dependence 161 References 163 5 Graphical Diagnostics, Tests, and Model Selection 167 5.1 Basic Graphical Diagnostics 5.2 Hypothesis Tests 173 5.2. 1 Tests of Independence 173 5.2.2 Tests of Exchangeability ∴.176 5.2.3 A Test of Radial symmetr .178 5.2.4 Tests of Extreme-Value Dependence 17 5.2.5 Goodness-of-Fit Tests l81 5.2.6 A Mixture of Graphical and Formal Goodness-of-Fit ests 188 5.3 Model selection 191 References 195 6 Ties, Time Series, and regression 197 6.1①Ti ··: 198 6.2 Selected Copula Tests and Models for Time Series 216 6.2. 1 Tests of Stationarity ∴.216 6.2.3 Models for Multivariate Time Series Based on∴……….226 6.2.2 Tests of Serial Independence Conditional Copulas… 230 6.3 Regression.…………… 238 References 252 a R and Package versions........................... 255 Index 259

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