Introduction to Bayesian estimation and copula models of dependence / Arkady Shemyakin, Alexander Kniazev.

Format:

Book

Author/Creator:

Shemyakin, Arkady, author.

Kniazev, Alexander (Mathematician), author.

Series:

THEi Wiley ebooks.

THEi Wiley ebooks

Language:

English

Subjects (All):

Bayesian statistical decision theory.

Copulas (Mathematical statistics).

Physical Description:

1 online resource (349 pages) : illustrations (some color)

Edition:

1st edition

Place of Publication:

Hoboken, New Jersey : John Wiley & Sons, Incorporated, 2017.

System Details:

Access using campus network via VPN at home (THEi Users Only).

text file

Summary:

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. In addition, this book includes: • 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-l...

Contents:

Intro

Introduction to Bayesian Estimation and Copula Models of Dependence

Contents

List of Figures

List of Tables

Acknowledgments

Acronyms

Glossary

About the Companion Website

Introduction

Part I Bayesian Estimation

1 Random Variables and Distributions

1.1 Conditional Probability

1.2 Discrete Random Variables

1.3 Continuous Distributions on the Real Line

1.4 Continuous Distributions with Nonnegative Values

1.5 Continuous Distributions on a Bounded Interval

1.6 Joint Distributions

1.7 Time-Dependent Random Variables

References

2 Foundations of Bayesian Analysis

2.1 Education and Wages

2.2 Two Envelopes

2.3 Hypothesis Testing

2.3.1 The Likelihood Principle

2.3.2 Review of Classical Procedures

2.3.3 Bayesian Hypotheses Testing

2.4 Parametric Estimation

2.4.1 Review of Classical Procedures

2.4.2 Maximum Likelihood Estimation

2.4.3 Bayesian Approach to Parametric Estimation

2.5 Bayesian and Classical Approaches to Statistics

2.5.1 Classical (Frequentist) Approach

2.5.2 Lady Tasting Tea

2.5.3 Bayes Theorem

2.5.4 Main Principles of the Bayesian Approach

2.6 The Choice of the Prior

2.6.1 Subjective Priors

2.6.2 Objective Priors

2.6.3 Empirical Bayes

2.7 Conjugate Distributions

2.7.1 Exponential Family

2.7.2 Poisson Likelihood

2.7.3 Table of Conjugate Distributions

3 Background for Markov Chain Monte Carlo

3.1 Randomization

3.1.1 Rolling Dice

3.1.2 Two Envelopes Revisited

3.2 Random Number Generation

3.2.1 Pseudo-random Numbers

3.2.2 Inverse Transform Method

3.2.3 General Transformation Methods

3.2.4 Accept-Reject Methods

3.3 Monte Carlo Integration

3.3.1 Numerical Integration

3.3.2 Estimating Moments

3.3.3 Estimating Probabilities

3.3.4 Simulating Multiple Futures.

3.4 Precision of Monte Carlo Method

3.4.1 Monitoring Mean and Variance

3.4.2 Importance Sampling

3.4.3 Correlated Samples

3.4.4 Variance Reduction Methods

3.5 Markov Chains

3.5.1 Markov Processes

3.5.2 Discrete Time, Discrete State Space

3.5.3 Transition Probability

3.5.4 "Sun City"

3.5.5 Utility Bills

3.5.6 Classification of States

3.5.7 Stationary Distribution

3.5.8 Reversibility Condition

3.5.9 Markov Chains with Continuous State Spaces

3.6 Simulation of a Markov Chain

3.7 Applications

3.7.1 Bank Sizes

3.7.2 Related Failures of Car Parts

4 Markov Chain Monte Carlo Methods

4.1 Markov Chain Simulations for Sun City and Ten Coins

4.2 Metropolis-Hastings Algorithm

4.3 Random Walk MHA

4.4 Gibbs Sampling

4.5 Diagnostics of MCMC

4.5.1 Monitoring Bias and Variance of MCMC

4.5.2 Burn-in and Skip Intervals

4.5.3 Diagnostics of MCMC

4.6 Suppressing Bias and Variance

4.6.1 Perfect Sampling

4.6.2 Adaptive MHA

4.6.3 ABC and Other Methods

4.7 Time-to-Default Analysis of Mortgage Portfolios

4.7.1 Mortgage Defaults

4.7.2 Customer Retention and Infinite Mixture Models

4.7.3 Latent Classes and Finite Mixture Models

4.7.4 Maximum Likelihood Estimation

4.7.5 A Bayesian Model

PART II Modeling Dependence

5 Statistical Dependence Structures

5.1 Introduction

5.2 Correlation

5.2.1 Pearson's Linear Correlation

5.2.2 Spearman's Rank Correlation

5.2.3 Kendall's Concordance

5.3 Regression Models

5.3.1 Heteroskedasticity

5.3.2 Nonlinear Regression

5.3.3 Prediction

5.4 Bayesian Regression

5.5 Survival Analysis

5.5.1 Proportional Hazards

5.5.2 Shared Frailty

5.5.3 Multistage Models of Dependence

5.6 Modeling Joint Distributions

5.6.1 Bivariate Survival Functions

5.6.2 Bivariate Normal.

5.6.3 Simulation of Bivariate Normal

5.7 Statistical Dependence and Financial Risks

5.7.1 A Story of Three Loans

5.7.2 Independent Defaults

5.7.3 Correlated Defaults

6 Copula Models of Dependence

6.1 Introduction

6.2 Definitions

6.2.1 Quasi-Monotonicity

6.2.2 Definition of Copula

6.2.3 Sklar's Theorem

6.2.4 Survival Copulas

6.3 Simplest Pair Copulas

6.3.1 Maximum Copula

6.3.2 Minimum Copula

6.3.3 FGM Copulas

6.4 Elliptical Copulas

6.4.1 Elliptical Distributions

6.4.2 Method of Inverses

6.4.3 Gaussian Copula

6.4.4 The t-copula

6.5 Archimedean Copulas

6.5.1 Definitions

6.5.2 One-Parameter Copulas

6.5.3 Clayton Copula

6.5.4 Frank Copula

6.5.5 Gumbel-Hougaard Copula

6.5.6 Two-Parameter Copulas

6.6 Simulation of Joint Distributions

6.6.1 Bivariate Elliptical Distributions

6.6.2 Bivariate Archimedean Copulas

6.7 Multidimensional Copulas

7 Statistics of Copulas

7.1 The Formula that Killed Wall Street

7.2 Criteria of Model Comparison

7.2.1 Goodness-of-Fit Tests

7.2.2 Posterior Predictive p-values

7.2.3 Information Criteria

7.2.4 Concordance Measures

7.2.5 Tail Dependence

7.3 Parametric Estimation

7.3.1 Parametric, Semiparametric, or Nonparametric?

7.3.2 Method of Moments

7.3.3 Minimum Distance

7.3.4 MLE and MPLE

7.3.5 Bayesian Estimation

7.4 Model Selection

7.4.1 Hybrid Approach

7.4.2 Information Criteria

7.4.3 Bayesian Model Selection

7.5 Copula Models of Joint Survival

7.6 Related Failures of Vehicle Components

7.6.1 Estimation of Association Parameters

7.6.2 Comparison of Copula Classes

7.6.3 Bayesian Model Selection

7.6.4 Conclusions

8 International Markets

8.1 Introduction

8.2 Selection of Univariate Distribution Models.

8.3 Prior Elicitation for Pair Copula Parameter

8.4 Bayesian Estimation of Pair Copula Parameters

8.5 Selection of Pair Copula Model

8.6 Goodness-of-Fit Testing

8.7 Simulation and Forecasting

Index

EULA.

Notes:

Includes bibliographical references and index.

Description based on print version record.

ISBN:

9781118959022

1118959027

9781118959046

1118959043

OCLC:

974040698