Elements of Copula Modeling with R

ISBN-13: 9783319896342 / Angielski / Miękka / 2019 / 267 str.

Marius Hofert; Ivan Kojadinovic; Martin Machler

Opis

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).

Spis treści

Preface 5

1 Introduction 9

1.1 A motivating example . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2 Probability and quantile transformations . . . . . . . . . . . . . . . . . 11

1.3 Copulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.4 Structure and philosophy of the book . . . . . . . . . . . . . . . . . . . 14

1.5 Additional references . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2 Copulas 17

2.1 Denition and characterization . . . . . . . . . . . . . . . . . . . . . . 17

2.2 The Frechet{Hoeding bounds . . . . . . . . . . . . . . . . . . . . . . . 27

2.3 Sklar's Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.4 The invariance principle . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.5 Survival copulas and copula symmetries . . . . . . . . . . . . . . . . . 49

2.6 Measures of association . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.6.1 Fallacies related to the correlation coecient . . . . . . . . . . . 55

2.6.2 Rank correlation measures . . . . . . . . . . . . . . . . . . . . . 60

2.6.3 Tail dependence coecients . . . . . . . . . . . . . . . . . . . . . 67

2.7 Rosenblatt transform and conditional sampling . . . . . . . . . . . . . 76

3 Classes and families 87

3.1 Elliptical distributions and copulas . . . . . . . . . . . . . . . . . . . . 87

3.1.1 Elliptical distributions . . . . . . . . . . . . . . . . . . . . . . . . 87

3.1.2 Elliptical copulas . . . . . . . . . . . . . . . . . . . . . . . . . . 92

3.2 Archimedean copulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

3.3 Extreme-value copulas . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

3.4 Selected copula transformations and constructions . . . . . . . . . . . . 122

3.4.1 Rotated copulas . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

3.4.2 Khoudraji's device . . . . . . . . . . . . . . . . . . . . . . . . . . 126

3.4.3 Mixtures of copulas . . . . . . . . . . . . . . . . . . . . . . . . . 132

4 Estimation 137

4.1 Estimation under a parametric assumption on the copula . . . . . . . . 137

4.1.1 Parametrically estimated margins . . . . . . . . . . . . . . . . . 138

4.1.2 Non-parametrically estimated margins . . . . . . . . . . . . . . . 142

4.1.3 Estimators of elliptical copula parameters . . . . . . . . . . . . . 151

4.1.4 Other semi-parametric estimators . . . . . . . . . . . . . . . . . 156

4.1.5 Estimation of copula models with partly xed parameters . . . . 156

4.2 Non-parametric estimation of the copula . . . . . . . . . . . . . . . . . 161

4.2.1 The empirical copula . . . . . . . . . . . . . . . . . . . . . . . . 161

4.2.2 Under extreme-value dependence . . . . . . . . . . . . . . . . . . 164

5 Graphical diagnostics, tests and model selection 167

5.1 Basic graphical diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . 167

5.2 Hypothesis tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

5.2.1 Tests of independence . . . . . . . . . . . . . . . . . . . . . . . . 173

5.2.2 Tests of exchangeability . . . . . . . . . . . . . . . . . . . . . . . 175

5.2.3 A test of radial symmetry . . . . . . . . . . . . . . . . . . . . . . 177

5.2.4 Tests of extreme-value dependence . . . . . . . . . . . . . . . . . 178

5.2.5 Goodness-of-t tests . . . . . . . . . . . . . . . . . . . . . . . . . 181

5.2.6 A mixture of graphical and formal goodness-of-t tests . . . . . 188

5.3 Model selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

6 Ties, time series and regression 195

6.1 Ties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

6.2 Selected copula tests and models for time series . . . . . . . . . . . . . 214

6.2.1 Tests of stationarity . . . . . . . . . . . . . . . . . . . . . . . . . 214

6.2.2 Tests of serial independence . . . . . . . . . . . . . . . . . . . . 225

6.2.3 Models for multivariate time series based on conditional copulas 229

6.3 Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

A R and package versions 251

References 255

Index 267

O autorze

The four authors of the book are the authors of the R package copula available on CRAN.  

Marius Hofert is an assistant professor of statistics at the University of Waterloo, Canada. He obtained his Ph.D. in mathematics from the University of Ulm, Germany in 2010. He then held a postdoctoral research position at ETH Zurich, Switzerland. After guest assistant professorships at the Technical University Munich, Germany and the University of Washington, USA, he joined the Department of Statistics and Actuarial Science at the University of Waterloo in 2014. His main research interests lie in copula modeling, computational statistics, data science and quantitative risk management.

Ivan Kojadinovic is a professor of statistics at the University of Pau, France. He received his Ph.D. from the University of Reunion, France in 2002 and joined the University of Nantes, France in 2003 as an assistant professor. From 2007 to 2010, he was a lecturer and then a  senior lecturer at the Department of Statistics of the University of Auckland, New Zealand, before joining the University of Pau in 2010. His research interests lie in nonparametric statistics, copulas, change-point detection, and environmental and financial applications.

Martin Mächler is a lecturer and senior scientist at the ETH Zurich, Switzerland. He received his Ph.D. in mathematics from the ETH in 1989, and spent his postdoc years at the University of Washington, Seattle and Bell Communications Research (Bellcore), before joining the Seminar für Statistik at the ETH as lecturer in 1991. He became involved with R in 1995, was a founding member of the R core team in 1997 and has since been active in the development of R. His research interests include nonparametric curve estimation, numerical approximation, clustering, robust statistics, sparse matrices and statistical computing in general. He has been the maintainer of circa 20 CRAN R packages, including the "recommended" packages Matrix and cluster.

Jun Yan is a professor of statistics at the University of Connecticut, USA. He received his Ph.D. in statistics from University of Wisconsin - Madison, USA in 2003. He was an assistant professor at the University of Iowa, USA before joining UConn in 2007. His research interests include survival analysis, clustered data analysis, multivariate dependence, spatial extremes, and statistical computing. Actively involved in collaborative research in public health and environmental sciences, he has a special interest in making advanced statistical methods widely accessible via open source software.

Blurb

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.

In the spirit of the Use R! series, each chapter combines key theoretical definitions or results with illustrations in R. Aimed at statisticians, actuaries, risk managers, engineers and environmental scientists wanting to learn about the theory and practice of copula modeling using R without an overwhelming amount of mathematics, the book can also be used for teaching a course on copula modeling.