# Introduction to Bayesian Estimation and Copula Models of Dependence

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Presents an introduction to Bayesian statistics, presents an emphasis on Bayesian methods (prior and posterior), Bayes estimation, prediction, MCMC,Bayesian regression, and Bayesian analysis of statistical modelsof dependence, and features a focus on copulas for risk management

Introduction to Bayesian Estimation and Copula Models of Dependence emphasizes the applications of Bayesian analysis to copula modeling and equips readers with the tools needed to implement the procedures of Bayesian estimation in copula models of dependence. This book is structured in two parts: the first four chapters serve as a general introduction to Bayesian statistics with a clear emphasis on parametric estimation and the following four chapters stress statistical models of dependence with a focus of copulas.

A review of the main concepts is discussed along with the basics of Bayesian statistics including prior information and experimental data, prior and posterior distributions, with an emphasis on Bayesian parametric estimation. The basic mathematical background of both Markov chains and Monte Carlo integration and simulation is also provided. The authors discuss statistical models of dependence with a focus on copulas and present a brief survey of pre-copula dependence models. The main definitions and notations of copula models are summarized followed by discussions of real-world cases that address particular risk management problems.

• Practical examples of copulas in use including within the Basel Accord II documents that regulate the world banking system as well as examples of Bayesian methods within current FDA recommendations

• Step-by-step procedures of multivariate data analysis and copula modeling, allowing readers to gain insight for their own applied research and studies

• Separate reference lists within each chapter and end-of-the-chapter exercises within Chapters 2 through 8

• A companion website containing appendices: data files and demo files in Microsoft® Office Excel®, basic code in R, and selected exercise solutions

Introduction to Bayesian Estimation and Copula Models of Dependence is a reference and resource for statisticians who need to learn formal Bayesian analysis as well as professionals within analytical and risk management departments of banks and insurance companies who are involved in quantitative analysis and forecasting. This book can also be used as a textbook for upper-undergraduate and graduate-level courses in Bayesian statistics and analysis.

ARKADY SHEMYAKIN, PhD, is Professor in the Department of Mathematics and Director of the Statistics Program at the University of St. Thomas. A member of the American Statistical Association and the International Society for Bayesian Analysis, Dr. Shemyakin's research interests include informationtheory, Bayesian methods of parametric estimation, and copula models in actuarial mathematics, finance, and engineering.

ALEXANDER KNIAZEV, PhD, is Associate Professor and Head of the Department of Mathematics at Astrakhan State University in Russia. Dr. Kniazev's research interests include representation theory of Lie algebras and finite groups, mathematical statistics, econometrics, and financial mathematics.

List of Figures xiii

List of Tables xvii

Acknowledgments xix

Acronyms xxi

Glossary xxiii

Introduction xxvii

Part I Bayesian Estimation 1

1. Random Variables and Distributions 3

1.1 Conditional Probability 3

1.2 Discrete Random Variables 5

1.3 Continuous Distributions on the Real Line 8

1.4 Continuous Distributions with Nonnegative Values 13

1.5 Continuous Distributions on a Bounded Interval 19

1.6 Joint Distributions 20

1.7 Time-Dependent Random Variables 27

References 32

2. Foundations of Bayesian Analysis 33

2.1 Education andWages 33

2.2 Two Envelopes 36

2.3 Hypothesis Testing 39

2.3.1 The Likelihood Principle 39

2.3.2 Review of Classical Procedures 40

2.3.3 Bayesian Hypotheses Testing 42

2.4 Parametric Estimation 43

2.4.1 Review of Classical Procedures 43

2.4.2 Maximum Likelihood Estimation 44

2.4.3 Bayesian Approach to Parametric Estimation 46

2.5 Bayesian and Classical Approaches to Statistics 47

2.5.1 Classical (Frequentist) Approach 49

2.5.3 Bayes Theorem 53

2.5.4 Main Principles of the Bayesian Approach 55

2.6 The Choice of the Prior 57

2.6.1 Subjective Priors 57

2.6.2 Objective Priors 60

2.6.3 Empirical Bayes 63

2.7 Conjugate Distributions 66

2.7.1 Exponential Family 66

2.7.2 Poisson Likelihood 67

2.7.3 Table of Conjugate Distributions 68

References 68

Exercises 69

3. Background for Markov Chain Monte Carlo 73

3.1 Randomization 73

3.1.1 Rolling Dice 73

3.1.2 Two Envelopes Revisited 74

3.2 Random Number Generation 76

3.2.1 Pseudo-random Numbers 76

3.2.2 Inverse Transform Method 77

3.2.3 General Transformation Methods 78

3.2.4 Accept–Reject Methods 81

3.3 Monte Carlo Integration 85

3.3.1 Numerical Integration 86

3.3.2 Estimating Moments 87

3.3.3 Estimating Probabilities 88

3.3.4 Simulating Multiple Futures 90

3.4 Precision of Monte Carlo Method 91

3.4.1 Monitoring Mean and Variance 91

3.4.2 Importance Sampling 94

3.4.3 Correlated Samples 96

3.4.4 Variance Reduction Methods 97

3.5 Markov Chains 101

3.5.1 Markov Processes 101

3.5.2 Discrete Time, Discrete State Space 103

3.5.3 Transition Probability 103

3.5.4 “Sun City” 104

3.5.5 Utility Bills 105

3.5.6 Classification of States 105

3.5.7 Stationary Distribution 107

3.5.8 Reversibility Condition 108

3.5.9 Markov Chains with Continuous State Spaces 108

3.6 Simulation of a Markov Chain 109

3.7 Applications 111

3.7.1 Bank Sizes 111

3.7.2 Related Failures of Car Parts 113

References 115

Exercises 117

4. Markov Chain Monte Carlo Methods 119

4.1 Markov Chain Simulations for Sun City and Ten Coins 119

4.2 Metropolis–Hastings Algorithm 126

4.3 Random Walk MHA 130

4.4 Gibbs Sampling 134

4.5 Diagnostics of MCMC 136

4.5.1 Monitoring Bias and Variance of MCMC 137

4.5.2 Burn-in and Skip Intervals 140

4.5.3 Diagnostics of MCMC 142

4.6 Suppressing Bias and Variance 144

4.6.1 Perfect Sampling 144

4.6.3 ABC and Other Methods 145

4.7 Time-to-Default Analysis of Mortgage Portfolios 146

4.7.1 Mortgage Defaults 146

4.7.2 Customer Retention and Infinite Mixture Models 147

4.7.3 Latent Classes and Finite Mixture Models 149

4.7.4 Maximum Likelihood Estimation 150

4.7.5 A Bayesian Model 151

References 156

Exercises 158

Part II Modeling Dependence 159

5. Statistical Dependence Structures 161

5.1 Introduction 161

5.2 Correlation 165

5.2.1 Pearson’s Linear Correlation 165

5.2.2 Spearman’s Rank Correlation 167

5.2.3 Kendall’s Concordance 167

5.3 Regression Models 170

5.3.1 Heteroskedasticity 171

5.3.2 Nonlinear Regression 172

5.3.3 Prediction 175

5.4 Bayesian Regression 176

5.5 Survival Analysis 179

5.5.1 Proportional Hazards 180

5.5.2 Shared Frailty 180

5.5.3 Multistage Models of Dependence 182

5.6 Modeling Joint Distributions 182

5.6.1 Bivariate Survival Functions 183

5.6.2 Bivariate Normal 185

5.6.3 Simulation of Bivariate Normal 185

5.7 Statistical Dependence and Financial Risks 186

5.7.1 A Story of Three Loans 186

5.7.2 Independent Defaults 188

5.7.3 Correlated Defaults 189

References 192

Exercises 193

6. Copula Models of Dependence 195

6.1 Introduction 195

6.2 Definitions 196

6.2.1 Quasi-Monotonicity 197

6.2.2 Definition of Copula 198

6.2.3 Sklar’s Theorem 198

6.2.4 Survival Copulas 199

6.3 Simplest Pair Copulas 200

6.3.1 Maximum Copula 200

6.3.2 Minimum Copula 202

6.3.3 FGM Copulas 203

6.4 Elliptical Copulas 204

6.4.1 Elliptical Distributions 204

6.4.2 Method of Inverses 205

6.4.3 Gaussian Copula 205

6.4.4 The t-copula 207

6.5 Archimedean Copulas 209

6.5.1 Definitions 209

6.5.2 One-Parameter Copulas 210

6.5.3 Clayton Copula 212

6.5.4 Frank Copula 213

6.5.5 Gumbel–Hougaard Copula 214

6.5.6 Two-Parameter Copulas 216

6.6 Simulation of Joint Distributions 217

6.6.1 Bivariate Elliptical Distributions 218

6.6.2 Bivariate Archimedean Copulas 219

6.7 Multidimensional Copulas 222

References 228

Exercises 230

7. Statistics of Copulas 233

7.1 The Formula that Killed Wall Street 233

7.2 Criteria of Model Comparison 237

7.2.1 Goodness-of-Fit Tests 237

7.2.2 Posterior Predictive p-values 239

7.2.3 Information Criteria 241

7.2.4 Concordance Measures 243

7.2.5 Tail Dependence 244

7.3 Parametric Estimation 245

7.3.1 Parametric, Semiparametric, or Nonparametric? 246

7.3.2 Method of Moments 248

7.3.3 Minimum Distance 248

7.3.4 MLE and MPLE 249

7.3.5 Bayesian Estimation 250

7.4 Model Selection 252

7.4.1 Hybrid Approach 252

7.4.2 Information Criteria 254

7.4.3 Bayesian Model Selection 256

7.5 Copula Models of Joint Survival 257

7.6 Related Failures of Vehicle Components 260

7.6.1 Estimation of Association Parameters 261

7.6.2 Comparison of Copula Classes 262

7.6.3 Bayesian Model Selection 265

7.6.4 Conclusions 267

References 268

Exercises 271

8. International Markets 273

8.1 Introduction 273

8.2 Selection of Univariate Distribution Models 276

8.3 Prior Elicitation for Pair Copula Parameter 280

8.4 Bayesian Estimation of Pair Copula Parameters 286

8.5 Selection of Pair Copula Model 290

8.6 Goodness-of-Fit Testing 295

8.7 Simulation and Forecasting 298

References 304

Exercises 307

Index 309

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