Dependence Modeling with Copulas  book cover
1st Edition

Dependence Modeling with Copulas

ISBN 9781466583221
Published June 26, 2014 by Chapman and Hall/CRC
480 Pages 21 B/W Illustrations

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Book Description

Dependence Modeling with Copulas covers the substantial advances that have taken place in the field during the last 15 years, including vine copula modeling of high-dimensional data. Vine copula models are constructed from a sequence of bivariate copulas. The book develops generalizations of vine copula models, including common and structured factor models that extend from the Gaussian assumption to copulas. It also discusses other multivariate constructions and parametric copula families that have different tail properties and presents extensive material on dependence and tail properties to assist in copula model selection.

The author shows how numerical methods and algorithms for inference and simulation are important in high-dimensional copula applications. He presents the algorithms as pseudocode, illustrating their implementation for high-dimensional copula models. He also incorporates results to determine dependence and tail properties of multivariate distributions for future constructions of copula models.

Table of Contents

Dependence modeling
Early research for multivariate non-Gaussian
Copula representation for a multivariate distribution
Data examples: scatterplots and semi-correlations
Likelihood analysis and model comparisons
Copula models versus alternative multivariate models
Terminology for multivariate distributions with U(0, 1) margins
Copula constructions and properties

Basics: Dependence, Tail Behavior, and Asymmetries
Multivariate cdfs and their conditional distributions
Laplace transforms
Extreme value theory
Tail heaviness
Probability integral transform
Multivariate Gaussian/normal
Elliptical and multivariate t distributions
Multivariate dependence concepts
Fréchet classes and Fréchet bounds, given univariate margins
Fréchet classes given higher order margins
Concordance and other dependence orderings
Measures of bivariate monotone association
Tail dependence
Tail asymmetry
Measures of bivariate asymmetry
Tail order
Semi-correlations of normal scores for a bivariate copula
Tail dependence functions
Strength of dependence in tails and boundary conditional cdfs
Conditional tail expectation for bivariate distributions
Tail comonotonicity
Summary for analysis of properties of copulas

Copula Construction Methods
Overview of dependence structures and desirable properties
Archimedean copulas based on frailty/resilience
Archimedean copulas based on Williamson transform
Hierarchical Archimedean and dependence
Mixtures of max-id
Another limit for max-id distributions
Fréchet class given bivariate margins
Mixtures of conditional distributions
Vine copulas or pair-copula constructions
Factor copula models
Combining models for different groups of variables
Nonlinear structural equation models
Truncated vines, factor models and graphical models
Copulas for stationary time series models
Multivariate extreme value distributions
Multivariate extreme value distributions with factor structure
Other multivariate models
Operations to get additional copulas
Summary for construction methods

Parametric Copula Families and Properties
Summary of parametric copula families
Properties of classes of bivariate copulas
Copulas based on the logarithmic series LT
Copulas based on the gamma LT
Copulas based on the Sibuya LT
Copulas based on the positive stable LT
Galambos extreme value
Hüsler-Reiss extreme value
Archimedean with LT that is integral of positive stable
Archimedean based on LT of inverse gamma
Multivariate tv
Marshall-Olkin multivariate exponential
Asymmetric Gumbel/Galambos copulas
Extreme value limit of multivariate tv
Copulas based on the gamma stopped positive stable LT
Copulas based on the gamma stopped gamma LT
Copulas based on the positive stable stopped gamma LT
Gamma power mixture of Galambos
Positive stable power mixture of Galambos
Copulas based on the Sibuya stopped positive stable LT
Copulas based on the Sibuya stopped gamma LT
Copulas based on the LT of generalized Sibuya
Copulas based on the tilted positive stable LT
Copulas based on the shifted negative binomial LT
Multivariate GB2 distribution and copula
Factor models based on convolution-closed families
Morgenstern or FGM
Fréchet’s convex combination
Additional parametric copula families
Dependence comparisons
Summary for parametric copula families

Inference, Diagnostics, and Model Selection
Parametric inference for copulas
Likelihood inference
Log-likelihood for copula models
Maximum likelihood: asymptotic theory
Inference functions and estimating equations
Composite likelihood
Kullback-Leibler divergence
Initial data analysis for copula models
Copula pseudo likelihood, sensitivity analysis
Non-parametric inference
Diagnostics for conditional dependence
Diagnostics for adequacy of fit
Vuong’s procedure for parametric model comparisons
Summary for inference

Computing and Algorithms
Roots of nonlinear equations
Numerical optimization and maximum likelihood
Numerical integration and quadrature
Numerical methods involving matrices
Graphs and spanning trees
Computation of τ, ρS, and ρN for copulas
Computation of empirical Kendall’s τ
Simulation from multivariate distributions and copulas
Likelihood for vine copula
Likelihood for factor copula
Copula derivatives for factor and vine copulas
Generation of vines
Simulation from vines and truncated vine models
Partial correlations and vines
Partial correlations and factor structure
Searching for good truncated R-vine approximations
Summary for algorithms

Applications and Data Examples
Data analysis with misspecified copula models
Inferences on tail quantities
Discretized multivariate Gaussian and R-vine approximation
Insurance losses: bivariate continuous
Longitudinal count: multivariate discrete
Count time series
Multivariate extreme values
Multivariate financial returns
Conservative tail inference
Item response: multivariate ordinal
SEM model as vine: alienation data
SEM model as vine: attitude-behavior data
Overview of applications

Theorems for Properties of Copulas
Absolutely continuous and singular components of multivariate distributions
Continuity properties of copulas
Dependence concepts
Fréchet classes and compatibility
Archimedean copulas
Multivariate extreme value distributions
Mixtures of max-id distributions
Elliptical distributions
Tail dependence
Tail order
Combinatorics of vines
Vines and mixtures of conditional distributions
Factor copulas
Kendall functions
Laplace transforms
Regular variation
Summary for further research

Appendix: Laplace Transforms and Archimedean Generators


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"This monograph is an essential compendium for any researcher working with copulas, and I am sure that it will become the primary reference for anything ‘copula’. It is mathematically rigorous with consistent notation and attention to detail in every each chapter, researchers will find a comprehensive and precise description of a specific research topic with many invaluable references to relevant recent publications...A must-have on the bookshelf of any statistician interested in multivariate modelling!"
Australian and New Zealand Journal of Statistics, March 2016

"… a ‘must have’ for someone seriously involved in dependence modeling with copulas, especially with a focus on modeling real data. The huge collection of facts and references for certain families of copulas, dependence measures, and statistical tools makes this book a valuable reference for researchers and experienced practitioners. I expect the statistical approach to the field will be especially appealing for the JASA audience."
Journal of the American Statistical Association, December 2015

"Harry Joe’s impressive new book Dependence Modeling with Copulas will undoubtedly become a key reference work in the field. … this excellent book will be a welcome addition to the library of anyone with an interest in copulas, multivariate statistics, or models of dependence. The researcher will find the book indispensable while the applied statistician will find much of value to guide the choice of copula models in data analysis. The book is packed with information … an interested reader will return to the text again and again, making new discoveries each time."
Journal of Time Series Analysis, 2015