Loss Models: From Data to Decisions

702 Pages·2012·15.98 MB·English

by  Klugman

Contents
Half Title page
Title page
Copyright page
Preface
Part I: Introduction
Chapter 1: Modeling
1.1 The model-based approach
1.2 Organization of this book
Chapter 2: Random Variables
2.1 Introduction
2.2 Key functions and four models
Chapter 3: Basic Distributional Quantities
3.1 Moments
3.2 Percentiles
3.3 Generating functions and sums of random variables
3.4 Tails of distributions
3.5 Measures of Risk
Part II: Actuarial Models
Chapter 4: Characteristics of Actuarial Models
4.1 Introduction
4.2 The role of parameters
Chapter 5: Continuous Models
5.1 Introduction
5.2 Creating new distributions
5.3 Selected distributions and their relationships
5.4 The linear exponential family
Chapter 6: Discrete Distributions
6.1 Introduction
6.2 The Poisson distribution
6.3 The negative binomial distribution
6.4 The binomial distribution
6.5 The (a, b, 0) class
6.6 Truncation and modification at zero
Chapter 7: Advanced Discrete Distributions
7.1 Compound frequency distributions
7.2 Further properties of the compound Poisson class
7.3 Mixed frequency distributions
7.4 Effect of exposure on frequency
Appendix: An inventory of discrete distributions
Chapter 8: Frequency and Severity with Coverage Modifications
8.1 Introduction
8.2 Deductibles
8.3 The loss elimination ratio and the effect of inflation for ordinary
deductibles
8.4 Policy limits
8.5 Coinsurance, deductibles, and limits
8.6 The impact of deductibles on claim frequency
Chapter 9: Aggregate Loss Models
9.1 Introduction
9.2 Model choices
9.3 The compound model for aggregate claims
9.4 Analytic results
9.5 Computing the aggregate claims distribution
9.6 The recursive method
9.7 The impact of individual policy modifications on aggregate payments
9.8 The individual risk model
Part III: Construction of Empirical Models
Chapter 10: Review of Mathematical Statistics
10.1 Introduction
10.2 Point estimation
10.3 Interval estimation
10.4 Tests of hypotheses
Chapter 11: Estimation for Complete Data
11.1 Introduction
11.2 The empirical distribution for complete, individual data
11.3 Empirical distributions for grouped data
Chapter 12: Estimation for Modified Data
12.1 Point estimation
12.2 Means, variances, and interval estimation
12.3 Kernel density models
12.4 Approximations for large data sets
Part IV: Parametric Statistical Methods
Chapter 13: Frequentist Estimation
13.1 Method of moments and percentile matching
13.2 Maximum likelihood estimation
13.3 Variance and interval estimation
13.4 Nonnormal confidence intervals
13.5 Maximum likelihood estimation of decrement probabilities
Chapter 14: Frequentist Estimation for Discrete Distributions
14.1 Poisson
14.2 Negative binomial
14.3 Binomial
14.4 The (a, b,1) class
14.5 Compound models
14.6 Effect of exposure on maximum likelihood estimation
14.7 Exercises
Chapter 15: Bayesian Estimation
15.1 Definitions and Bayes’ Theorem
15.2 Inference and prediction
15.3 Conjugate prior distributions and the linear exponential family
15.4 Computational issues
Chapter 16: Model Selection
16.1 Introduction
16.2 Representations of the data and model
16.3 Graphical comparison of the density and distribution functions
16.4 Hypothesis tests
16.5 Selecting a model
Part V: Credibility
Chapter 17: Introduction and Limited Fluctuation Credibility
17.1 Introduction
17.2 Limited fluctuation credibility theory
17.3 Full credibility
17.4 Partial credibility
17.5 Problems with the approach
17.6 Notes and References
17.7 Exercises
Chapter 18: Greatest Accuracy Credibility
18.1 introduction
18.2 Conditional distributions and expectation
18.3 The Bayesian methodology
18.4 The credibility premium
18.5 The Bühlmann model
18.6 The Bühlmann–Straub model
18.7 Exact credibility
18.8 Notes and References
18.9 Exercises
Chapter 19: Empirical Bayes Parameter Estimation
19.1 Introduction
19.2 Nonparametric estimation
19.3 Semi parametric estimation
19.4 Notes and References
19.5 Exercises
Part VI: Simulation
Chapter 20: Simulation
20.1 Basics of simulation
20.2 Simulation for specific distributions
20.3 Determining the sample size
20.4 Examples of simulation in actuarial modeling
Appendix A: An Inventory of Continuous Distributions
A.1 Introduction
A.2 Transformed beta family
A.3 Transformed gamma family
A.4 Distributions for large losses
A.5 Other distributions
A.6 Distributions with finite support
Appendix B: An Inventory of Discrete Distributions
B.1 Introduction
B.2 The (a, b, 0) class
B.3 The (a, b, 1) class
B.4 The compound class
B.5 A hierarchy of discrete distributions
Appendix C: Frequency and Severity Relationships
Appendix D: The Recursive Formula
Appendix E: Discretization of the Severity Distribution
E.1 The method of rounding
E.2 Mean preserving
E.3 Undiscretization of a discretized distribution
Appendix F: Numerical Optimization and Solution of Systems of Equations
F.1 Maximization using Solver
F.2 The simplex method
F.3 Using Excel® to solve equations
References
Index
LOSS MODELS
WILEY SERIES IN PROBABILITY AND STATISTICS
ESTABLISHED BY WALTER A. SHEWHART AND SAMUEL S. WILKS
Editors: David J. Balding, Noel A. C. Cressie, Garrett M. Fitzmaurice, Harvey
Goldstein, Iain M. Johnstone, Geert Molenberghs, David W. Scott, Adrian F.
M, Smith, Ruey S. Tsay, Sanford Weisberg Editors Emeriti: Vic Barnett, J.
Stuart Hunter, Joseph B. Kadane, Jozef L. Teugels The Wiley Series in
Probability and Statistics is well established and authoritative. It covers many
topics of current research interest in both pure and applied statistics and
probability theory. Written by leading statisticians and institutions, the titles
span both state-of-the-art developments in the field and classical methods.
Reflecting the wide range of current research in statistics, the series
encompasses applied, methodological and theoretical statistics, ranging from
applications and new techniques made possible by advances in computerized
practice to rigorous treatment of theoretical approaches.
This series provides essential and invaluable reading for all statisticians,
whether in academia, industry, government, or research.
† ABRAHAM and LEDOLTER · Statistical Methods for Forecasting
AGRESTI · Analysis of Ordinal Categorical Data, Second Edition
AGRESTI · An Introduction to Categorical Data Analysis, Second Edition
AGRESTI · Categorical Data Analysis, Second Edition
ALTMAN, GILL, and McDONALD · Numerical Issues in Statistical
Computing for the Social Scientist AMARATUNGA and CABRERA ·
Exploration and Analysis of DNA Microarray and Protein Array Data AND L ·
Mathematics of Chance ANDERSON · An Introduction to Multivariate
Statistical Analysis, Third Edition * ANDERSON · The Statistical Analysis of
Time Series ANDERSON, AUQUIER, HAUCK, OAKES, VANDAELE, and
WEISBERG · Statistical Methods for Comparative Studies ANDERSON and
LOYNES · The Teaching of Practical Statistics
ARMITAGE and DAVID (editors) · Advances in Biometry
ARNOLD, BALAKRISHNAN, and NAGARAJA · Records
* ARTHANARI and DODGE · Mathematical Programming in Statistics *
BAILEY · The Elements of Stochastic Processes with Applications to the
Natural Sciences BAJORSKI · Statistics for Imaging, Optics, and Photonics
BALAKRISHNAN and KOUTRAS · Runs and Scans with Applications
BALAKRISHNAN and NG · Precedence-Type Tests and Applications
BARNETT · Comparative Statistical Inference, Third Edition
BARNETT · Environmental Statistics
BARNETT and LEWIS · Outliers in Statistical Data, Third Edition
BARTHOLOMEW, KNOTT, and MOUSTAKI · Latent Variable Models and
Factor Analysis: A Unified Approach, Third Edition BARTOSZYNSKI and
NIEWIADOMSKA-BUGAJ · Probability and Statistical Inference, Second
Edition BASILEVSKY · Statistical Factor Analysis and Related Methods:
Theory and Applications BATES and WATTS · Nonlinear Regression Analysis
and Its Applications
BECHHOFER, SANTNER, and GOLDSMAN · Design and Analysis of
Experiments for Statistical Selection, Screening, and Multiple Comparisons
BEIRLANT, GOEGEBEUR, SEGERS, TEUGELS, and DE WAAL · Statistics
of Extremes: Theory and Applications BELSLEY · Conditioning Diagnostics:
Collinearity and Weak Data in Regression
† BELSLEY, KUH, and WELSCH · Regression Diagnostics: Identifying
Influential Data and Sources of Collinearity BENDAT and PIERSOL · Random
Data: Analysis and Measurement Procedures, Fourth Edition BERNARDO and
SMITH · Bayesian Theory
BERZUINI, DAWID, and BERNARDINELL · Causality: Statistical
Perspectives and Applications BHAT and MILLER · Elements of Applied
Stochastic Processes, Third Edition BHATTACHARYA and WAYMIRE ·
Stochastic Processes with Applications
BIEMER, GROVES, LYBERG, MATHIOWETZ, and SUDMAN ·
Measurement Errors in Surveys
BILLINGSLEY · Convergence of Probability Measures, Second Edition
BILLINGSLEY · Probability and Measure, Anniversary Edition
BIRKES and DODGE · Alternative Methods of Regression
BISGAARD and KULAHCI · Time Series Analysis and Forecasting by
Example
BISWAS, DATTA, FINE, and SEGAL · Statistical Advances in the
Biomedical Sciences: Clinical Trials, Epidemiology, Survival Analysis, and
Bioinformatics BLISCHKE and MURTHY (editors) · Case Studies in Reliability
and Maintenance
BLISCHKE and MURTHY · Reliability: Modeling, Prediction, and