Introduction to Bayesian Estimation and Copula Models of Dependence

Arkady Shemyakin, Alexander Kniazev

• Publisher: Wiley
• Year: 2017
• Print ISBN: 9781118959015
• eBook ISBN: 9781118959022
• Edition: 1
• Topic: Mathematik
• Subtopic: Wahrscheinlichkeitsrechnung & Statistiken

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-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.

Part I
Bayesian Estimation

1
Random Variables and Distributions

Chapter 1 is by no means suggested to replace or replicate a standard course in probability. Its purpose is to provide a reference source and remind the readers what topics they might need to review. For a systematic review of probability and introduction to statistics we can recommend excellent texts by DeGroot and Schervish [4], Miller and Miller [10], and Rice [12]. In-depth coverage of probability distributions in the context of loss models is offered by Klugman et al. in [8]. If the reader is interested in a review with a comprehensive software guide, we can recommend Crawley’s handbook in R [3].

Here we will introduce the main concepts and notations used throughout the book. The emphasis is made on the simplicity of explanations, and often in order to avoid technical details we have to sacrifice mathematical rigor and conciseness. We will also introduce a library of distributions for further illustrations. Without a detailed reference to the main facts of probability theory, we need to however emphasize the role played in the sequel by the concept of conditional probability, which becomes our starting point.

1.1 Conditional Probability

Let A and B be two random events, which could be represented as two subsets of the same sample space S including all possible outcomes of a chance experiment: A⊆S and B⊆S. Conditional probability of B given A measures the chances of B to happen if A is already known to occur. It can be defined for events A and B such that P(A) > 0 as

(1.1)

where P(A∩B) = P(B∩A) is the probability of intersection of A and B, the event indicating that both A and B occur. This conditional probability should not be confused with the conditional probability of A given B defined as

(1.2)

which shares the same numerator with P(B∣A), but has a different denominator.

The source of possible confusion is a different choice of the “sample space” or “reference population”—whatever language one prefers to use—in (1.1) and (1.2) corresponding to the denominators in these formulas. In case of (1.1) we consider only such cases that A occurs, so that the sample space or reference population is reduced from S to A, while in (1.2) it changes from S to B.

To illustrate this distinction, we will use a simple example. It fits the purpose of the book, using many illustrations from the fields of insurance and risk management, to begin with an example related to insurance.

In a fictitious country of Endolacia, people drive cars and buy insurance against accidents. Accidents do happen on a regular basis, but not too often. During t...

Table of contents

• Cover
• Titlepage
• Copyright
• Dedication
• Acknowledgments
• Acronyms
• Glossary
• About the Companion Website
• Introduction
• Part I: Bayesian Estimation
• Part II: Modeling Dependence
• Index
• EULA