Introduction to Bayesian estimation and copula models of dependence. (2017)

• Record Type: Book
• Further Information: Note: Arkady Shemyakin, Alexander Kniazev.
• Authors: Shemyakin, Arkady; (Mathematician), Kniazev, Alexander
• Contents: List of Figures xv List of Tables xxi Foreword xxiii Acknowledgments xxv Acronyms xxvii Glossary xxix Introduction xxxi PART I BAYESAIN ESTIMATION 1 Random Variables and Distributions 3 1.1 Conditional Probability 4 1.2 Discrete Random Variables 5 1.3 Continuous Distributions on the Real Line 8 1.4 Continuous Distributions with Nonnegative Values 12 1.5 Continuous Distributions on a Bounded Interval 17 1.6 Joint Distributions 18 1.7 Time Dependent Random Variables 24 References 29 2 Foundations of Bayesian Analysis 31 2.1 Education and Wages 31 2.2 Two Envelopes 34 2.3 Hypothesis Testing 37 2.3.1 The Likelihood Principle. 37 2.3.2 Review of Classical Procedures 37 2.3.3 Bayesian Hypotheses Testing 39 2.4 Parametric Estimation 41 2.4.1 Review of Classical Procedures 41 2.4.2 Maximum Likelihood Estimation (MLE) 41 2.4.3 Bayesian Approach to Parametric Estimation 44 2.5 Bayesian and Classical Approaches to Statistics 45 2.5.1 Classical (Frequentist) Approach 47 2.5.2 Lady Tasting Tea 47 2.5.3 Bayes Theorem 50 2.5.4 Main Principles of the Bayesian Approach 51 2.6 The Choice of the Prior 54 2.6.1 Subjective Priors 54 2.6.2 Objective Priors 57 2.6.3 Empirical Bayes 60 2.7 Conjugate Distributions 62 2.7.1 Exponential Family 63 2.7.2 Poisson Likelihood 63 2.7.3 Table of Conjugate Distributions 64 References 64 Problems 66 3 Background for Markov Chain Monte Carlo 69 3.1 Randomization 69 3.1.1 Rolling Dice 69 3.1.2 Two Envelopes Revisited 70 3.2 Random Number Generation 72 3.2.1 Pseudorandom Numbers 72 3.2.2 Inverse Transform Method 73 3.2.3 General Transformation Methods 75 3.2.4 AcceptReject Methods 78 3.3 Monte Carlo Integration 81 3.3.1 Numerical Integration 82 3.3.2 Estimating Moments 83 3.3.3 Estimating Probabilities 84 3.3.4 Simulating Multiple Futures 85 3.4 Precision of Monte Carlo Method 87 3.4.1 Monitoring Mean and Variance 87 3.4.2 Importance Sampling 89 3.4.3 Correlated Samples 91 3.4.4 Variance Reduction Methods 93 3.5 Markov Chains 96 3.5.1 Markov Processes 96 3.5.2 Discrete Time, Discrete State Space 98 3.5.3 Transition Probability 98 3.5.4 "Sun City" 99 3.5.5 Utility Bills 99 3.5.6 Classification of States 100 3.5.7 Stationary Distribution 101 3.5.8 Reversibility Condition 102 3.5.9 Markov Chains with Continuous State Spaces 103 3.6 Simulation of a Markov Chain 104 3.7 Applications 105 3.7.1 Bank Sizes 105 3.7.2 Related Failures of Car Parts 107 References 110 Problems 111 4 Markov Chain Monte Carlo Methods (MCMC) 113 4.1 Markov Chain Simulations for Sun City and Ten Coins 113 4.2 MetropolisHastings Algorithm (MHA) 120 4.3 Random Walk MHA 123 4.4 Gibbs Sampling 126 4.5 Diagnostics of MCMC 129 4.5.1 Monitoring Bias and Variance of MCMC 130 4.5.2 Burnin and Skip Intervals 133 4.5.3 Diagnostics of MCMC 134 4.6 Suppressing Bias and Variance 135 4.6.1 Perfect Sampling 135 4.6.2 Adaptive MHA 136 4.6.3 ABC and Other Methods 137 4.7 Timetodefault Analysis of Mortgage Portfolios 137 4.7.1 Mortgage Defaults 137 4.7.2 Customer Retention and Infinite Mixture Models 138 4.7.3 Latent Classes and Finite Mixture Models 140 4.7.4 Maximum Likelihood Estimation 142 4.7.5 A Bayesian Model 143 References 146 Problems 148 PART II MODELING DEPENDENCE 5 Statistical Dependence Structures 153 5.1 Introduction 153 5.2 Correlation 156 5.2.1 Pearson’s Linear Correlation 157 5.2.2 Spearman’s Rank Correlation 158 5.2.3 Kendall’s Concordance 159 5.3 Regression Models 161 5.3.1 Heteroskedasticity 162 5.3.2 Nonlinear Regression 163 5.3.3 Prediction 166 5.4 Bayesian Regression 167 5.5 Survival Analysis 170 5.5.1 Proportional Hazards 171 5.5.2 Shared Frailty 171 5.5.3 Multistage Models of Dependence 173 5.6 Modeling Joint Distributions 173 5.6.1 Bivariate Survival Functions 174 5.6.2 Bivariate Normal 176 5.7 Statistical Dependence and Financial Risks 177 5.7.1 A Story of Three Loans 177 5.7.2 Independent Defaults 179 5.7.3 Correlated Defaults 180 References 183 Problems 184 6 Copula Models of Dependence 187 6.1 Introduction 187 6.2 Definitions 189 6.2.1 Quasimonotonicity 189 6.2.2 Definition of Copula 190 6.2.3 Sklar’s Theorem 191 6.2.4 Survival Copulas 191 6.3 Simplest Pair Copulas 192 6.3.1 Maximum Copula 192 6.3.2 Minimum Copula 195 6.3.3 FGM Copulas 196 6.4 Elliptical Copulas 197 6.4.1 Elliptical Distributions 197 6.4.2 Method of Inverses 198 6.4.3 Gaussian Copula 198 6.4.4 The tcopula 200 6.5 Archimedean Copulas 202 6.5.1 Definitions 202 6.5.2 Oneparameter Copulas 204 6.5.3 Clayton Copula 205 6.5.4 Frank Copula 206 6.5.5 GumbelHougaard Copula 207 6.5.6 Twoparameter Copulas 209 6.6 Simulation of Joint Distributions 211 6.6.1 Bivariate Elliptical Distributions 211 6.6.2 Bivariate Archimedean Copulas 213 6.7 Multidimensional Copulas 215 References 222 Problems 224 7 Statistics of Copulas 227 7.1 The Formula that Killed Wall Street 227 7.2 Criteria of Model Comparison 231 7.2.1 Goodnessoffit Tests 232 7.2.2 Posterior Predictive pvalues 233 7.2.3 Information Criteria 236 7.2.4 Concordance Measures 237 7.2.5 Tail Dependence 238 7.3 Parametric Estimation 239 7.3.1 Parametric, Semiparametric, or Nonparametric? 241 7.3.2 Method of Moments 242 7.3.3 Minimum Distance 242 7.3.4 MLE and MPLE 243 7.3.5 Bayesian Estimation 244 7.4 Model Selection 246 7.4.1 Hybrid Approach 246 7.4.2 Information Criteria 247 7.4.3 Bayesian Model Selection 249 7.5 Copula Models of Joint Survival 250 7.6 Related Failures of Vehicle Components 253 7.6.1 Estimation of Association Parameters 255 7.6.2 Comparison of Copula Classes 256 7.6.3 Bayesian Model Selection 258 7.6.4 Conclusions 260 References 262 Problems 264 8 International Markets 265 8.1 Introduction 265 8.2 Selection of univariate distribution models. 268 8.3 Prior Elicitation for Pair Copula Parameter 272 8.4 Bayesian Estimation of Pair Copula Parameters 277 8.5 Selection of Pair Copula Model 281 8.6 GoodnessofFit Testing 285 8.7 Simulation and Forecasting 289 References 294 Problems 296 Index 299
• Edition: 1st
• Publisher Details: Hoboken, New Jersey : John Wiley & Sons, Inc
• Publication Date: 2017
• Extent: 1 online resource
• Subjects: 519.542; Bayesian statistical decision theory; Copulas (Mathematical statistics)
• Languages: English
• ISBNs: 9781118959022; 9781118959039
• Related ISBNs: 9781118959015
• Notes: Note: Description based on CIP data; item not viewed.
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