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E-raamat: Dependence Modeling with Copulas

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Dependence Modeling with CopulasSuurem pilt
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This is not a revision of his 1997 book, says Joe, but an updated expansion of a few chapters in it into a full account of copula theory, inference, and application. The vice copula construction has been a big advance for copula modeling with high-dimensional data, he explains, and as a class, vine copula models are constructed from a sequence of bivariate copulas, some applied to pairs of univariate margins and other applied to pairs of univariate conditional distributions. He covers basics: dependence, tail behavior, and asymmetries; copula construction methods; parametric copula families and properties; inference, diagnostics, and model selection; computing and algorithms; applications and data examples; and theorems for properties of copulas. Annotation ©2014 Ringgold, Inc., Portland, OR (protoview.com)

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.

Arvustused

"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 Joes 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

Preface xiii
Notation xv
1 Introduction
1(24)
1.1 Dependence modeling
1(1)
1.2 Early research for multivariate non-Gaussian
2(5)
1.3 Copula representation for a multivariate distribution
7(2)
1.4 Data examples: scatterplots and semi-correlations
9(6)
1.5 Likelihood analysis and model comparisons
15(8)
1.5.1 A brief summary of maximum likelihood
16(1)
1.5.2 Two-stage estimation for copula models
16(1)
1.5.3 Likelihood analysis for continuous data: insurance loss
17(2)
1.5.4 Likelihood analysis for discrete data: ordinal response
19(4)
1.6 Copula models versus alternative multivariate models
23(1)
1.7 Terminology for multivariate distributions with U(0, 1) margins
23(1)
1.8 Copula constructions and properties
24(1)
2 Basics: dependence, tail behavior and asymmetries
25(60)
2.1 Multivariate cdfs and their conditional distributions
26(7)
2.1.1 Conditions for multivariate cdfs
26(2)
2.1.2 Absolutely continuous and singular components
28(1)
2.1.3 Conditional cdfs
29(2)
2.1.4 Mixture models and conditional independence models
31(1)
2.1.5 Power of a cdf or survival function
32(1)
2.2 Laplace transforms
33(1)
2.3 Extreme value theory
34(2)
2.4 Tail heaviness
36(2)
2.5 Probability integral transform
38(1)
2.6 Multivariate Gaussian/normal
38(3)
2.7 Elliptical and multivariate T distributions
41(4)
2.8 Multivariate dependence concepts
45(2)
2.8.1 Positive quadrant and orthant dependence
45(1)
2.8.2 Stochastically increasing positive dependence
46(1)
2.8.3 Right-tail increasing and left-tail decreasing
46(1)
2.8.4 Associated random variables
46(1)
2.8.5 Total positivity of order 2
47(1)
2.9 Frechet classes and Frechet bounds, given univariate margins
47(3)
2.10 Frechet classes given higher order margins
50(1)
2.11 Concordance and other dependence orderings
51(2)
2.12 Measures of bivariate monotone association
53(9)
2.12.1 Kendall's tau
55(1)
2.12.2 Spearman's rank correlation
56(1)
2.12.3 Blomqvist's beta
57(1)
2.12.4 Correlation of normal scores
58(1)
2.12.5 Auxiliary results for dependence measures
58(1)
2.12.6 Magnitude of asymptotic variance of measures of associations
59(1)
2.12.7 Measures of association for discrete/ordinal variables
60(2)
2.13 Tail dependence
62(2)
2.14 Tail asymmetry
64(1)
2.15 Measures of bivariate asymmetry
65(2)
2.16 Tail order
67(3)
2.16.1 Tail order function and copula density
68(2)
2.17 Semi-correlations of normal scores for a bivariate copula
70(3)
2.18 Tail dependence functions
73(7)
2.19 Strength of dependence in tails and boundary conditional cdfs
80(1)
2.20 Conditional tail expectation for bivariate distributions
80(4)
2.21 Tail comonotonicity
84(1)
2.22 Summary for analysis of properties of copulas
84(1)
3 Copula construction methods
85(74)
3.1 Overview of dependence structures and desirable properties
86(3)
3.2 Archimedean copulas based on frailty/resilience
89(4)
3.3 Archimedean copulas based on Williamson transform
93(2)
3.4 Hierarchical Archimedean and dependence
95(3)
3.5 Mixtures of max-id
98(4)
3.6 Another limit for max-id distributions
102(4)
3.7 Frechet class given bivariate margins
106(1)
3.8 Mixtures of conditional distributions
106(1)
3.9 Vine copulas or pair-copula constructions
107(21)
3.9.1 Vine sequence of conditional distributions
108(2)
3.9.2 Vines as graphical models
110(2)
3.9.3 Vine distribution: conditional distributions and joint density
112(2)
3.9.4 Vine array
114(1)
3.9.5 Vines with some or all discrete marginal distributions
115(2)
3.9.6 Truncated vines
117(1)
3.9.7 Multivariate distributions for which the simplifying assumption holds
118(8)
3.9.8 Vine equivalence classes
126(1)
3.9.9 Historical background of vines
127(1)
3.10 Factor copula models
128(8)
3.10.1 Continuous response
128(4)
3.10.2 Discrete ordinal response
132(3)
3.10.3 Mixed continuous and ordinal response
135(1)
3.10.4 t-Copula with factor correlation matrix structure
135(1)
3.11 Combining models for different groups of variables
136(4)
3.11.1 Bi-factor copula model
137(1)
3.11.2 Nested dependent latent variables
137(1)
3.11.3 Dependent clusters with conditional independence
138(2)
3.12 Nonlinear structural equation models
140(3)
3.13 Truncated vines, factor models and graphical models
143(1)
3.14 Copulas for stationary time series models
144(4)
3.15 Multivariate extreme value distributions
148(2)
3.16 Multivariate extreme value distributions with factor structure
150(1)
3.17 Other multivariate models
151(4)
3.17.1 Analogy of Archimedean and elliptical
152(1)
3.17.2 Other constructions
153(2)
3.18 Operations to get additional copulas
155(3)
3.19 Summary for construction methods
158(1)
4 Parametric copula families and properties
159(64)
4.1 Summary of parametric copula families
160(2)
4.2 Properties of classes of bivariate copulas
162(1)
4.3 Gaussian
163(1)
4.3.1 Bivariate Gaussian copula
163(1)
4.3.2 Multivariate Gaussian copula
164(1)
4.4 Plackett
164(1)
4.5 Copulas based on the logarithmic series LT
165(3)
4.5.1 Bivariate Frank
165(1)
4.5.2 Multivariate Frank extensions
166(2)
4.6 Copulas based on the gamma LT
168(2)
4.6.1 Bivariate Mardia-Takahasi-Clayton-Cook-Johnson
168(1)
4.6.2 Multivariate MTCJ extensions
168(2)
4.7 Copulas based on the Sibuya LT
170(1)
4.7.1 Bivariate Joe/B5
170(1)
4.7.2 Multivariate extensions with Sibuya LT
171(1)
4.8 Copulas based on the positive stable LT
171(3)
4.8.1 Bivariate Gumbel
171(1)
4.8.2 Multivariate Gumbel extensions
172(2)
4.9 Galambos extreme value
174(1)
4.9.1 Bivariate Galambos copula
174(1)
4.9.2 Multivariate Galambos extensions
175(1)
4.10 Husler-Reiss extreme value
175(2)
4.10.1 Bivariate Husler-Reiss
176(1)
4.10.2 Multivariate Husler-Reiss
176(1)
4.11 Archimedean with LT that is integral of positive stable
177(3)
4.11.1 Bivariate copula in Joe and Ma, 2000
177(3)
4.11.2 Multivariate extension
180(1)
4.12 Archimedean based on LT of inverse gamma
180(1)
4.13 Multivariate tv
181(1)
4.14 Marshall-Olkin multivariate exponential
182(3)
4.14.1 Bivariate Marshall-Olkin
182(2)
4.14.2 Multivariate Marshall-Olkin exponential and extensions
184(1)
4.15 Asymmetric Gumbel/Galambos copulas
185(4)
4.15.1 Asymmetric Gumbel with Marshall-Olkin at boundary
185(1)
4.15.2 Asymmetric Gumbel based on deHaan representation
186(3)
4.16 Extreme value limit of multivariate tv
189(1)
4.16.1 Bivariate t-EV
189(1)
4.17 Copulas based on the gamma stopped positive stable LT
190(3)
4.17.1 Bivariate BB1: Joe and Hu, 1996
190(2)
4.17.2 BB1: range of pairs of dependence measures
192(1)
4.17.3 Multivariate extensions of BB1
193(1)
4.18 Copulas based on the gamma stopped gamma LT
193(2)
4.18.1 Bivariate BB2: Joe and Hu, 1996
193(2)
4.19 Copulas based on the positive stable stopped gamma LT
195(1)
4.19.1 Bivariate BB3: Joe and Hu, 1996
195(1)
4.20 Gamma power mixture of Galambos
196(3)
4.20.1 Bivariate BB4: Joe and Hu, 1996
197(1)
4.20.2 Multivariate extensions of BB4
198(1)
4.21 Positive stable power mixture of Galambos
199(1)
4.21.1 Bivariate BB5: Joe and Hu, 1996
199(1)
4.22 Copulas based on the Sibuya stopped positive stable LT
200(1)
4.22.1 Bivariate BB6: Joe and Hu, 1996
200(1)
4.23 Copulas based on the Sibuya stopped gamma LT
201(2)
4.23.1 Bivariate BB7: Joe and Hu, 1996
202(1)
4.24 Copulas based on the generalized Sibuya LT
203(2)
4.24.1 Bivariate BB8; Joe 1993
204(1)
4.25 Copulas based on the tilted positive stable LT
205(1)
4.25.1 Bivariate BB9 or Crowder
205(1)
4.26 Copulas based on the shifted negative binomial LT
206(2)
4.26.1 Bivariate BB10
206(2)
4.27 Multivariate GB2 distribution and copula
208(2)
4.28 Factor models based on convolution-closed families
210(2)
4.29 Morgenstern or FGM
212(2)
4.29.1 Bivariate FGM
213(1)
4.29.2 Multivariate extensions of FGM
213(1)
4.30 Frechet's convex combination
214(1)
4.31 Additional parametric copula families
214(6)
4.31.1 Archimedean copula: LT is integral of Mittag-Leffler LT
215(1)
4.31.2 Archimedean copula based on positive stable stopped Sibuya LT
216(1)
4.31.3 Archimedean copula based on gamma stopped Sibuya LT
216(1)
4.31.4 3-parameter families with a power parameter
217(3)
4.32 Dependence comparisons
220(2)
4.33 Summary for parametric copula families
222(1)
5 Inference, diagnostics and model selection
223(36)
5.1 Parametric inference for copulas
223(2)
5.2 Likelihood inference
225(1)
5.3 Log-likelihood for copula models
226(1)
5.4 Maximum likelihood: asymptotic theory
227(1)
5.5 Inference functions and estimating equations
228(4)
5.5.1 Resampling methods for interval estimates
231(1)
5.6 Composite likelihood
232(2)
5.7 Kullback-Leibler divergence
234(9)
5.7.1 Sample size to distinguish two densities
235(1)
5.7.2 Jeffreys' divergence and KL sample size
236(4)
5.7.3 Kullback-Leibler divergence and maximum likelihood
240(2)
5.7.4 Discretized multivariate Gaussian
242(1)
5.8 Initial data analysis for copula models
243(3)
5.8.1 Univariate models
244(1)
5.8.2 Dependence structure
245(1)
5.8.3 Joint tails
246(1)
5.9 Copula pseudo likelihood, sensitivity analysis
246(1)
5.10 Non-parametric inference
247(4)
5.10.1 Empirical copula
247(1)
5.10.2 Estimation of functionals of a copula
248(2)
5.10.3 Non-parametric estimation of low-dimensional copula
250(1)
5.11 Diagnostics for conditional dependence
251(3)
5.12 Diagnostics for adequacy of fit
254(3)
5.12.1 Continuous variables
255(1)
5.12.2 Multivariate discrete and ordinal categorical
256(1)
5.13 Vuong's procedure for parametric model comparisons
257(1)
5.14 Summary for inference
258(1)
6 Computing and algorithms
259(50)
6.1 Roots of nonlinear equations
260(1)
6.2 Numerical optimization and maximum likelihood
261(1)
6.3 Numerical integration and quadrature
262(2)
6.4 Interpolation
264(1)
6.5 Numerical methods involving matrices
265(1)
6.6 Graphs and spanning trees
266(1)
6.7 Computation of τ, ρs and ρN for copulas
267(2)
6.8 Computation of empirical Kendall's τ
269(1)
6.9 Simulation from multivariate distributions and copulas
270(4)
6.9.1 Conditional method or Rosenblatt transform
270(1)
6.9.2 Simulation with reflected uniform random variables
271(1)
6.9.3 Simulation from product of cdfs
272(1)
6.9.4 Simulation from Archimedean copulas
272(1)
6.9.5 Simulation from mixture of max-id
273(1)
6.9.6 Simulation from multivariate extreme value copulas
274(1)
6.10 Likelihood for vine copula
274(5)
6.11 Likelihood for factor copula
279(2)
6.12 Copula derivatives for factor and vine copulas
281(6)
6.13 Generation of vines
287(3)
6.14 Simulation from vines and truncated vine models
290(7)
6.14.1 Simulation from vine copulas
291(2)
6.14.2 Simulation from truncated vines and factor copulas
293(4)
6.15 Partial correlations and vines
297(5)
6.16 Partial correlations and factor structure
302(1)
6.17 Searching for good truncated R-vine approximations
303(5)
6.17.1 Greedy sequential approach using minimum spanning trees
305(2)
6.17.2 Non-greedy algorithm
307(1)
6.18 Summary for algorithms
308(1)
7 Applications and data examples
309(54)
7.1 Data analysis with misspecified copula models
309(6)
7.1.1 Inference for dependence measures
310(3)
7.1.2 Inference for tail-weighted dependence measures
313(2)
7.2 Inferences on tail quantities
315(2)
7.3 Discretized multivariate Gaussian and R-vine approximation
317(2)
7.4 Insurance losses: bivariate continuous
319(3)
7.5 Longitudinal count: multivariate discrete
322(5)
7.6 Count time series
327(4)
7.7 Multivariate extreme values
331(4)
7.8 Multivariate financial returns
335(15)
7.8.1 Copula-GARCH
335(2)
7.8.2 Market returns
337(5)
7.8.3 Stock returns over several sectors
342(8)
7.9 Conservative tail inference
350(3)
7.10 Item response: multivariate ordinal
353(2)
7.11 SEM model as vine: alienation data
355(4)
7.12 SEM model as vine: attitude-behavior data
359(2)
7.13 Overview of applications
361(2)
8 Theorems for properties of copulas
363(66)
8.1 Absolutely continuous and singular components of multivariate distributions
363(2)
8.2 Continuity properties of copulas
365(1)
8.3 Dependence concepts
366(3)
8.4 Frechet classes and compatibility
369(5)
8.5 Archimedean copulas
374(8)
8.6 Multivariate extreme value distributions
382(4)
8.7 Mixtures of max-id distributions
386(5)
8.8 Elliptical distributions
391(3)
8.9 Tail dependence
394(4)
8.10 Tail order
398(2)
8.11 Combinatorics of vines
400(3)
8.12 Vines and mixtures of conditional distributions
403(7)
8.13 Factor copulas
410(9)
8.14 Kendall functions
419(3)
8.15 Laplace transforms
422(4)
8.16 Regular variation
426(1)
8.17 Summary for further reseach
427(2)
A Laplace transforms and Archimedean generators
429(8)
A.1 Parametric Laplace transform families
429(6)
A.1.1 One-parameter LT families
429(2)
A.1.2 Two-parameter LT families: group 1
431(2)
A.1.3 Two-parameter LT families: group 2
433(1)
A.1.4 LT families via integration
434(1)
A.2 Archimedean generators in Nelsen's book
435(2)
Bibliography 437(22)
Index 459
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