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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
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.
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Introduction to Bayesian Estimation and Copula Models of Dependence
Arkady Shemyakin
University of St. Thomas
Alexander Kniazev
Astrakhan State University
This edition first published 2017 © 2017 John Wiley & Sons, Inc.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by law. Advice on how to obtain permission to reuse material from this title is available at http://www.wiley.com/go/permissions
The right of Arkady Shemyakin and Alexander Kniazev to be identified as the author(s) of this work has been asserted in accordance with law.
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Library of Congress Cataloging-in-Publication Data:
Names: Shemyakin, Arkady. | Kniazev, Alexander (Mathematician) Title: Introduction to Bayesian estimation and copula models of dependence / Arkady Shemyakin, Alexander Kniazev. Description: Hoboken, New Jersey : John Wiley & Sons, Inc., [2017] | Includes index. Identifiers: LCCN 2016042826 | ISBN 9781118959015 (cloth) | ISBN 9781118959022 (epub) Subjects: LCSH: Bayesian statistical decision theory. | Copulas (Mathematical statistics) Classification: LCC QA279.5 .S435 2017 | DDC 519.5/42–dc23 LC record available at https://lccn.loc.gov/2016042826
Cover image: hakkiarslan/Gettyimages Cover design: Wiley
To all our families: special thanks to
CONTENTS
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
2.4 Parametric Estimation
2.5 Bayesian and Classical Approaches to Statistics
2.6 The Choice of the Prior
2.7 Conjugate Distributions
Exercises
References
3: Background for Markov Chain Monte Carlo
3.1 Randomization
3.2 Random Number Generation
3.3 Monte Carlo Integration
3.4 Precision of Monte Carlo Method
3.5 Markov Chains
3.6 Simulation of a Markov Chain
3.7 Applications
Exercises
References
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
4.6 Suppressing Bias and Variance
4.7 Time-to-Default Analysis of Mortgage Portfolios
Exercises
References
Part II: Modeling Dependence
5: Statistical Dependence Structures
5.1 Introduction
5.2 Correlation
5.3 Regression Models
5.4 Bayesian Regression
5.5 Survival Analysis
5.6 Modeling Joint Distributions
5.7 Statistical Dependence and Financial Risks
Exercises
References
6: Copula Models of Dependence
6.1 Introduction
6.2 Definitions
6.3 Simplest Pair Copulas
6.4 Elliptical Copulas
6.5 Archimedean Copulas
6.6 Simulation of Joint Distributions
6.7 Multidimensional Copulas
Exercises
References
7: Statistics of Copulas
7.1 The Formula that Killed Wall Street
7.2 Criteria of Model Comparison
7.3 Parametric Estimation
7.4 Model Selection
7.5 Copula Models of Joint Survival
7.6 Related Failures of Vehicle Components
Exercises
References
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
Exercises
References
Index
EULA
List of Tables
Chapter 3
Table 3.1
Table 3.2
Table 3.3
Table 3.4
Table 3.5
Chapter 4
Table 4.1
Table 4.2
Chapter 5
Table 5.1
Chapter 6
Table 6.1
Table 6.2
Table 6.3
Table 6.4
Chapter 7
Table 7.1
Table 7.2
Table 7.3
Table 7.4
Table 7.5
Table 7.6
Table 7.7
Table 7.8
Table 7.9
Table 7.10
Table 7.11
Table 7.12
Chapter 8
Table 8.1
Table 8.2
Table 8.3
Table 8.4
Table 8.5
Guide
Cover
Table of Contents
Part I: Bayesian Estimation
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Acknowledgments
We express our sincere appreciation to all of our friends and colleagues who contributed to this book in many different ways. We are grateful for many discussions we had and many comments we received from Paul Alper, Susan Callaway, Oleg Lepekhin, Alex McNeil, Olga Demidova, and many others.
We deeply appreciate the assistance we have received with collecting supplementary materials and preparing files for the companion website from Laura Hanson, Kathryn McKee, Cheryl Heskin, Matthew Galloway, Shannon Currier, Natalie Vandeweghe, and Stephanie Fritz.
We are grateful to our collaborators on the research projects which became the foundation for the book: Heekyung Youn, Natalia Kangina, Alicia Johnson, Vadim Gordeev, Matthew Galloway, Ellen Klingner, Nicole Lenz, Nicole Lopez, Kelsie Sargent, and Katheryn Wifvat.
We want to thank all of those to whom the material in the book was presented over the years and who contributed with their thoughtful comments regarding both the form and the content of the book. Special thanks are due to Sarah Millholland, Valentin Artemiev, Yuri Chepasov, Anastasia Rozhkova, Ekaterina Savinkina, Alexander Zyrianov, Doug Swisher, Laura Fink, Eric Schlicht, and many other students and seminar participants.
We would like to acknowledge the staff of our schools, Astrakhan State University and the University of St. Thomas in Minnesota, for their support of our work and making this project possible.
We also would like to thank all the editorial staff of Wiley: Susanne Steitz-Filler, Sari Friedman, Amy Hendrickson, Divya Narayanan, and others who at various stages helped with the completion of the book.
We are also grateful to Ekaterina Kniazeva and Alexandra Savinkina for proof-reading the text and making suggestions which helped immensely in improving the style of the book and hopefully making it readable.
Finally, our deepest gratitude goes to all our families, without whose support and companionship the project would not be possible.
A.S. and A.K.
Acronyms
ABS
asset-backed security
AR
autoregressive
ARCH
autoregressive conditional heteroskedasticity
ARIMA
autoregressive integrated moving averages
BB1
Joe’s BB1
BG
beta-geometric mixture model
BUGS
Bayes using Gibbs sampler (software)
CDO
collateralized debt obligation
CDS
credit default swap
CFTP
coupling from the past
CLT
the Central Limit Theorem
CMLE
canonical maximum likelihood estimation (estimates)
c.d.f.
cumulative distribution function
ENIAC
electronic numerical integrator and computer
FGM
Farlie–Gumbel–Morgenstern copulas
FTD
first-to-default (swaps)
GARCH
generalized autoregressive conditional heteroskedasticity
HAC
hierarchical Archimedean copula
HKC
hierarchical Kendall copula
IFM
inference from margins
i.i.d.
independent identically distributed
IMA
independent Metropolis algorithm
ISBA
International Society for Bayesian Analysis
JAGS
just another Gibbs sampler (software)
LLN
the Law of Large Numbers
MANIAC
mathematical analyzer, numerator, integrator, and computer
MCMC
Markov chain Monte Carlo
MHA
Metropolis–Hastings algorithm
MLE
maximum likelihood estimation (estimates)
MPLE
maximum pseudo-likelihood estimation (estimates)
PCC
pair copula construction
PD
probability of default
p.d.f
probability density function
RLUF
Rodriguez Lallena and Ubeda Flores copulas
RTO
regression through the origin
RWMHA
random walk Metropolis–Hastings algorithm
SACF
sample autocorrelation function
TTD
time-to-default
TTF
time-to-failure
WG
Weibull-gamma model
WGM
Weibull-gamma mixture model
WS
Weibull segmentation model
Glossary
E
(
X
)
expected value (mean)
Var
(
X
)
variance
Bin
(
n
,
p
)
binomial distribution with
n
trials and success probability
p
Poiss
(λ)
Poisson distribution
NB
(
p
,
r
)
negative binomial distribution
N
(μ, σ
2
)
normal (Gaussian) distribution with mean μ and variance σ
2
T
(η)
(Student)
t
-distribution with η degrees of freedom
Gamma
(
z
)
gamma function
Gamma
(α, λ)
gamma distribution with shape α and rate λ
Beta
(α, β)
beta distribution
Cov
(
X
,
Y
)
covariance
ρ(
X
,
Y
)
correlation
MN
(μ,
Σ
)
multivariate normal distribution
L
(
x
; θ)
likelihood function with data
x
and parameter θ
Λ(
x
)
likelihood ratio
PG
(1,
r
)
polygamma function order
r
X
∼
F
random variable
X
has distribution
F
∝
proportional
⪯
less or equal in sense of a special natural ordering
About the Companion Website
This book is accompanied by a companion website:
http://www.wiley.com/go/shemyakin/bayesian_estimation
The website includes:
Solutions to selected exercises
Excel dataset
Excel simulation templates
Appendices for Chapter 8
Datasets and results
Code in R
Introduction
Why does anyone need another book in Bayesian statistics? It seems that there already exist a lot of resources for those interested in the topic. There are many excellent books covering specific aspects of Bayesian analysis or providing a wide and comprehensive background of the entire field: Berger, Bernardo and Smith, Gamerman and Freitas Lopes, Gelman et al., Robert and Cassella, and many others. Most of these books, though, will assume a certain mathematical and statistical background and would rather fit a reader’s profile of a graduate or advanced graduate level. Out of those aimed at a less sophisticated audiences, we would certainly recommend excellent books of William Bolstad, John Kruschke, and Peter Lee. There also exist some very good books on copulas: comprehensive coverage by Nelsen and Joe, and also more application-related Cherubini et al., Emrechts et al., and some others. However, instead of just referring to these works and returning to our extensive to-do lists, we decided to spend considerable amount of time and effort putting together another book—the book we presently offer to the reader.
The main reason for our endeavor is: we target a very specific audience, which as we believe is not sufficiently serviced yet with Bayesian literature. We communicate with members of this audience routinely in our day-to-day work, and we have not failed to register that just providing them with reading recommendations does not seem to satisfy their needs. Our perceived audience could be loosely divided into two groups. The first includes advanced undergraduate students of Statistics, who in all likelihood have already had some exposure to main probabilistic and statistical principles and concepts (most likely, in classical or “frequentist” setup), and may (as we probably all do) exhibit some traces of Bayesian philosophy as applicable to their everyday lives. But for them these two: statistical methods on one hand and Bayesian thinking on the other, belong to very different spheres and do not easily combine in their decision-making process.
The second group consists of practitioners of statistical methods, working in their everyday lives on specific problems requiring the use of advanced quantitative analysis. They may be aware of a Bayesian alternative to classical methods and find it vaguely attractive, but are not familiar enough with formal Bayesian analysis in order to put it to work. These practitioners populate analytical departments of banks, insurance companies, and other major businesses. In short, they might be involved in predictive modeling, quantitative forecasting, and statistical reporting which often directly call for Bayesian approach.
In the recent years, we have frequently encountered representatives of both groups described above as our collaborators, be it in undergraduate research or in applied consulting projects, or both at once (such things do happen). We have discovered a frequent need to provide to them a crash course in Bayesian methods: prior and posterior, Bayes estimation, prediction, MCMC, Bayesian regression and time series, Bayesian analysis of statistical dependence. From this environment we get the idea to concisely summarize the methodology we normally share with our collaborators in order to provide the framework for successful joint projects. Later on this idea transformed itself into a phantasy to write this book and hand it to these two segments of the audience as a potentially useful resource. This intention determines the content of the book and dictates the necessity to cover specific topics in specific order, trying also to avoid any unnecessary detail. That is why we do not include a serious introduction to probability and classical statistics (we believe that our audience has at least some formal knowledge of the main principles and facts in these fields). Instead, in Chapter 1 we just offer a review of the concepts we will eventually use. If this review happens to be insufficient to some readers, it will hopefully at least inspire them to hit the introductory books which will provide a more comprehensive coverage.
Chapter 2 deals with the basics of Bayesian statistics: prior information and experimental data, prior and posterior distributions, with emphasis on Bayesian parametric estimation, just barely touching Bayesian hypothesis testing. Some time is spend on addressing subjective versus objective Bayesian paradigms and brief discussion of noninformative priors. We spend just so much time with conjugate priors and analytical derivation of Bayes estimators that will give an idea of the scope and limitations of the analytical approach. It seems likely that most readers in their practical applications will require the use of MCMC—Markov chain Monte Carlo method—the most efficient tool in the hands of modern Bayesians. Therefore, Chapter 3 contains the basic mathematical background on both Markov chains and Monte Carlo integration and simulation. In our opinion, successful use of Markov chain Monte Carlo methods is heavily based on good understanding on these two components. Speaking of Monte Carlo methods, the central idea of variance reduction nicely transitions us to MCMC and its diagnostics. Equally important, both Markov chains and Monte Carlo methods have numerous important applications outside of Bayesian setting, and these applications will be discussed as examples.
Chapter 4 covers MCMC per se. It may look suspicious from traditional point of view that we do not particularly emphasize Gibbs sampling, rather deciding to dwell on Metropolis–Hastings algorithm in its two basic versions: independent Metropolis and random walk Metropolis-Hastings. In our opinion, this approach allows us to minimize the theoretical exposure and get close to the point using simple examples. Also, in more advanced examples at the end of the book, Gibbs sampling will rarely work without Metropolis. Another disclosure we have to make: there exists a huge library of MCMC computer programs, including those available online as freeware. All necessary references are given in the text, including OpenBUGs and several R packages. However, we also introduce some very rudimentary computer code which allows the readers to “get inside” the algorithms. This expresses the authors’ firm belief that do-it-yourself is often the best way if not to actually apply statistical computing, but at least to learn how to use it.
This might be a good time to explain the authors’ attitude to the use of computer software while reading the book. Clearly, working on the exercises, many readers would find it handy to use some computing tools, and many readers would like to use the software of their choice (be it SPSS, Matlab, Stata, or any other packages). We try to structure our exercises and text examples in a way that makes it as easy as possible. What we offer from our perspective, in addition to this possibility, is a number of illustrations containing code and/or outputs in Microsoft Excel, Mathematica, and R. Our choice of Mathematica for providing graphics is simply explained by the background of the audience of the short courses where the material of the book has been approbated. We found it hard to refuse to treat ourselves and the readers to nice Mathematica graphical tools. R might be the software of choice for modern statisticians. Therefore we find its use along with this book both handy and inevitable. We can take only limited responsibility for the readers’ introduction to R, restricting ourselves to specific uses of this language to accompany the book. There exist a number of excellent R tutorials, both in print and online, and all necessary references are provided. We can especially recommend the R book by Crawley, and a book by Robert and Casella.
The first four chapters form the first part of the book which can be used as a general introduction to Bayesian statistics with a clear emphasis on the parametric estimation. Now we need to explain what is included in the remaining chapters, and what the link is between the book’s two parts. Our world is a complicated one. Due to the recent progress of communication tools and the globalization of the world economy and information space, we humans are less and less like separate universes leading our independent lives (was it ever entirely true?) Our physical lives, our economical existences, our information fields become more and more interrelated. Many processes successfully modeled in the past by probability and statistical methods assuming independent behavior of the components, become more and more intertwined. Brownian motion is still around, as well as Newtonian mechanics is, but it often fails to serve as a good model for many complicated systems with component interactions. This explains an increased interest to model statistical dependence: be it dependence of physical lives in demography and biology, dependence of financial markets, or dependence between the components of a complex engineering system. Out of the many models of statistical dependence, copulas play a special role. They provide an attractive alternative to such traditional tools as correlation analysis or Cox’s proportional hazards. The key factor in the popularity of copulas in applications to risk management is the way they model entire joint distribution function and are not limited to its moments. This allows for the treatment of nonlinear dependence including joint tail dependence and going far beyond the standard analysis of correlation. The limitations of more traditional correlation-based approaches to modeling risks were felt around the world during the last financial crisis.
Chapter 5 is dedicated to the brief survey of pre-copula dependence models, providing necessary background for Chapter 6, where the main definitions and notations of copula models are summarized. Special attention is dedicated to a somewhat controversial problem of model selection. Here, due to a wide variety of points of view expressed in the modern literature, the authors have to narrow down the survey in sync with their (maybe subjective) preferences.
Two types of copulas most popular in applications: Gaussian copulas and copulas from Archimedean family (Clayton, Frank, Gumbel–Hougaard, and some others) are introduced and compared from model selection standpoint in Chapter 7. Suggested principles of model selection have to be illustrated by more than just short examples. This explains the emergence of the last two sections of Chapters 7 and 8, which contain some cases dealing with particular risk management problems. The choice of the cases has to do with the authors’ recent research and consulting experience. The purpose of this chapter is to provide the readers with an opportunity to follow the procedures of multivariate data analysis and copula modeling step-by-step enabling them to use these cases as either templates or insights for their own applied research studies. The authors do not take on the ambitious goal to review the state-of-the-art Bayesian statistics or copula models of dependence. The emphasis is clearly made on applications of Bayesian analysis to copula modeling, which are still remarkably rare due to the factors discussed above as well as possibly to some other reasons unknown to the authors. The main focus of the book is on equipping the readers with the tools allowing them to implement the procedures of Bayesian estimation in copula models of dependence. These procedures seem to provide a path (one of many) into the statistics of the near future. The omens which are hard to miss: copulas found their way into Basel Accord II documents regulating the world banking system, Bayesian methods are mentioned in recent FDA recommendations.
The material of the book was presented in various combinations as the content of special topic courses at the both schools where the authors teach: the University of St. Thomas in Minnesota, USA, and Astrakhan State University in Astrakhan, Russia. Additionally, parts of this material have been presented as topic modules in graduate programs at MCFAM—the Minnesota Center for Financial and Actuarial Mathematics at the University of Minnesota and short courses at the Research University High School of Economics in Moscow, Russia, and U.S. Bank in Minneapolis, MN.
We can recommend this book as a text for a full one-semester course for advanced undergraduates with some background in probability and statistics. Part I (Chapters 1–4) can be used separately as an introduction to Bayesian statistics, while Part II (Chapters 5–8) can be used separately as an introduction to copula modeling for students with some prior knowledge of Bayesian statistics. We can also suggest it to accompany topics courses for students in a wide range of graduate programs. Each chapter is equipped with a separate reference list and Chapters 2–8 by their own sets of end-of-the-chapter exercises. The companion website contains Appendices: data files and demo files in Microsoft Excel, some simple code in R, and selected exercise solutions.
