Dependence Modeling with Copulas: 1st Edition (Hardback) book cover

Dependence Modeling with Copulas

1st Edition

By Harry Joe

Chapman and Hall/CRC

480 pages | 21 B/W Illus.

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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.

Reviews

"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 respect…in 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 Copulaswill 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

Table of Contents

Introduction

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

Gaussian

Plackett

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

Interpolation

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

Index

About the Series

Chapman & Hall/CRC Monographs on Statistics and Applied Probability

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Subject Categories

BISAC Subject Codes/Headings:
MAT029000
MATHEMATICS / Probability & Statistics / General