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## Multivariate models and dependence concepts

Author: Harry, Joe Publisher: Chapman and Hall, 1997. ; CRC, 1997.Language: EnglishDescription: 399 p. ; 24 cm.ISBN: 0412073315Type of document: BookBibliography/Index: Includes bibliographical references and index
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Europe Campus
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Print QA278 .J64 1997
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001175623
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Includes bibliographical references and index

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Multivariate Models and Dependence Concepts Contents Preface 1 Introduction 1.1 Overview and background 1.2 Style and format 1.3 Notation, abbreviations and conventions 1.4 Conditions for multivariate distribution functions 1.4.1 Properties of a bivariate cdf F 1.4.2 Properties of a multivariate cdf F 1.5 Types of dependence 1.6 Copulas 1.7 View of statistical modelling 1.8 Bibliographic notes 1.9 Exercises 2 Basic concepts of dependence 2.1 Dependence properties and measures 2.1.1 Positive quadrant and orthant dependence ° 2.1.2 Stochastic increasing positive dependence 2.1.3 Right-tail increasing and left-tail decreasing 2.1.4 Associated random variables 2.1.5 Total positivity of order 2 2.1.6 Positive function dependence 2.1.7 Relationships among dependence properties 2.1.8 Max-infinite and min-infinite divisibility 2.1.9 Kendall's tau and Spearman's rho ° 2.1.10 Tail dependence ° 2.1.11 Examples 2.2 Dependence orderings 2.2.1 Concordance ordering ° 2.2.2 Axioms for a bivariate dependence ordering xv 1 2 5 7 11 11 11 12 12 16 17 17 19 20 20 21 22 22 23 25 25 30 31 33 33 35 36 38 x CONTENTS 2.2.3 Axioms for a multivariate dependence ordering 38 2.2.4 More SI bivariate ordering " 40 2.2.5 More TP 2 bivariate orderings * 44 2.2.6 Positive function dependence ordering * 46 2.2.7 Examples: bivariate 48 2.2.8 Examples: multivariate 50 2.3 Bibliographic notes 52 2.4 Exercises 53 2.5 Unsolved problems 56 3 Fréchet classes 3.1 .F(F1 , ... , Fm ) ° 3.2 .F(F12, F13) 3.3 ;(F12 , F3) 3.4 F(F12, Fis, F23) 3.4.1 Bounds 3.4.2 Uniqueness 3.4.3 Compatibility conditions 3.5 .F(F123, F124, F134, F334) · 3.6 .F(F;j,1<i< j< m)* 3.7 General .F(Fs : S E S*), S' C Sm * 3.8 Bibliographic notes 3.9 Exercises 3.10 Unsolved problems 57 58 65 66 68 69 72 , 75 78 79 80 80 81 82 83 4 Construction of multivariate distributions 4.1 Desirable properties of a multivariate model ° 84 4.2 Laplace transforms and mixtures of powers ° 85 4.2.1 Dependence properties * 89 4.2.2 Frailty and proportional hazards 97 4.3 Mixtures of max -id distributions 98 4.3.1 Max- infinite divisibility conditions 101 4.3.2 Dependence properties · 102 4.4 Generalizations of functional forms 108 4.5 Mixtures of conditional distributions 111 4.5.1 Dependence properties * 115 4.6 Convolution-closed infinitely divisible class 0 118 4.7 Multivariate distributions given bivariate margins 120 4.7.1 Regression with binary variables 121 4.7.2 Maximum entropy given bivariate margins * 123 4.8 Molenberghs and Lesaffre construction 124 4.9 Spherically symmetric families: univariate margins 128 . CONTENTS xi 4.10 Other approaches 4.11' Bibliographic notes 4.12 Exercises 4.13 Unsolved problems 5 Parametric families of copulas 5.1 Bivariate one-parameter families ° 5.2 Bivariate two-parameter families 5.3 Multivariate copulas with partial symmetry 5.4 Extensions to negative dependence * 5.5 Multivariate copulas with general dependence 5.6 Bibliographic notes 5.7 Exercises 5.8 Unsolved problems 6 Multivariate extreme value distributions 6.1 Background: univariate extremes 6.2 Multivariate extreme value theory 6.2.1 Dependence properties 6.2.2 Extreme value limit results 6.3 Parametric families 6.3.1 Dependence families 6.3.2 Other parametric families 6.4 Point process approach * 6.5 Choice models 6.6 Mixtures of MEV distributions * 6.7 Bibliographic notes 6.8 Exercises 6.9 Unsolved problems 7 Multivariate discrete distributions 7.1 Multivariate binary 7.1.1 Bivariate Bernoulli and binomial ° 7.1.2 General multivariate Bernoulli ° 7.1.3 Exchangeable mixture model ° 7.1.4 Extensions to include covariates 7.1.5 Other exchangeable models 7.1.6 Mixture models ° 7.1.7 Latent variable models ° 7.1.8 Random effects models 7.1.9 Other general dependence models 7.1.10 Comparisons 134 134 135 138 139 139 149 155 157 163 166 166 168 169 169 172 177 179 182 182 191 194 197 202 206 207 208 209 209 210 211 211 215 216 219 221 224 225 226 xii CONTENTS 7.2 Multivariate count 7.2.1 Background for univariate count data ° 7.2.2 Multivariate Poisson ° 7.2.3 Mixture models and overdispersed Poisson ° 7.2.4 Other models 7.3 Multivariate models for ordinal responses ° 7.4 Multivariate models for nominal responses 7.5 Bibliographic notes 7.6 Exercises 7.7 Unsolved problems 8 Multivariate models with serial dependence 8.1 Markov chain models 8.1.1 Stationary time series based on copulas ° 8.1.2 Binary time series 8.1.3 Categorical response 8.1.4 Extreme value behaviour 8.2 k-dependent time series models 8.2.1 1-dependent series associated with copulas 8.2.2 Higher-order copulas 8.2.3 1-dependent binary series 8.3 Latent variable models 8.4 Convolution-closed infinitely divisible class 8.4.1 Stationary AR(1) time series ° 8.4.2 Moving average models 8.4.3 Higher-order autoregressive models 8.4.4 Models for longitudinal data 8.4.5 Other non-normal time series models 8.5 Markov chains: dependence properties * 8.6 Bibliographic notes 8.7 Exercises 8.8 Unsolved problems 9 Models from given conditional distributions 9.1 Conditional specifications and compatibility conditions 9.2 Examples 9.2.1 Conditional exponential density 9.2.2 Conditional exponential families 9.2.3 Conditional binary: logistic regressions 9.2.4 Binary: other conditional models 9.3 Bibliographic notes 232 232 233 234 236 236 237 239 240 242 243 244 244 246 248 249 253 253 255 256 258 259 259 264 265 268 269 270 280 281 282 283 283 285 285 288 290 294 296 CONTENTS xiii 9.4 Exercises 9.5 Unsolved problems 10 Statistical inference and computation 10.1 Estimation from likelihoods of margins ° 10.1.1 Asymptotic covariance matrix 10.1.2 Efficiency 10.1.3 Estimation consistency 10.1.4 Examples 10.2 Extensions 10.2.1 Covariates 10.2.2 Parameters common to more than one margin 10.3 Choice and comparison of models 10.4 Inference for Markov chains 10.5 Comments on Bayesian methods 10.6 Numerical methods 10.7 Bibliographic notes 10.8 Exercises 10.9 Unsolved problems 11 Data analysis and comparison of models 11.1 Example with multivariate binary response data 11.2 Example with multivariate ordinal response data 11.3 Example with multivariate extremes 11.4 Example with longitudinal binary data 11.5 Example with longitudinal count data 11.6 Example of inference for serially correlated data 11.7 Discussion 11.8 Exercises Appendix A.1 Laplace transforms A.2 Other background results A.2.1 Types of distribution functions and densities A.2.2 Convex functions and inequalities A.2.3 Maximum entropy A.3 Bibliographic notes References Index 296 296 297 299 301 305 306 307 311 311 315 316 317 319 319 321 321 322 323 324 336 344 352 357 369 371 372 373 373 377 377 379 380 382 383 395

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