Dependence Modeling with Copulas

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 Frechet classes and Frechet bounds, given univariate margins Frechet 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 Frechet 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 Husler-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 Frechet'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 tau, rhoS, and rhoN for copulas Computation of empirical Kendall's tau 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 Frechet 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