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Part V: Credibility
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Part V: Credibility
by Gordon E. Willmot, Harry H. Panjer, Stuart A. Klugman
Loss Models, 5th Edition
Cover
Series Page
Title Page
Copyright
Preface
About the Companion Website
Part I: Introduction
Chapter 1: Modeling
1.1 The Model-Based Approach
1.2 The 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 The Effect of Exposure on Frequency
7.5 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: Mathematical Statistics
Chapter 10: Introduction to Mathematical Statistics
10.1 Introduction and Four Data Sets
10.2 Point Estimation
10.3 Interval Estimation
10.4 The Construction of Parametric Estimators
10.5 Tests of Hypotheses
Chapter 11: Maximum Likelihood Estimation
11.1 Introduction
11.2 Individual Data
11.3 Grouped Data
11.4 Truncated or Censored Data
11.5 Variance and Interval Estimation for Maximum Likelihood Estimators
11.6 Functions of Asymptotically Normal Estimators
11.7 Nonnormal Confidence Intervals
Chapter 12: Frequentist Estimation for Discrete Distributions
12.1 The Poisson Distribution
12.2 The Negative Binomial Distribution
12.3 The Binomial Distribution
12.4 The (a,b,1) Class
12.5 Compound Models
12.6 The Effect of Exposure on Maximum Likelihood Estimation
12.7 Exercises
Chapter 13: Bayesian Estimation
13.1 Definitions and Bayes' Theorem
13.2 Inference and Prediction
13.3 Conjugate Prior Distributions and the Linear Exponential Family
13.4 Computational Issues
Part IV: Construction of Models
Chapter 14: Construction of Empirical Models
14.1 The Empirical Distribution
14.2 Empirical Distributions for Grouped Data
14.3 Empirical Estimation with Right Censored Data
14.4 Empirical Estimation of Moments
14.5 Empirical Estimation with Left Truncated Data
14.6 Kernel Density Models
14.7 Approximations for Large Data Sets
14.8 Maximum Likelihood Estimation of Decrement Probabilities
14.9 Estimation of Transition Intensities
Chapter 15: Model Selection
15.1 Introduction
15.2 Representations of the Data and Model
15.3 Graphical Comparison of the Density and Distribution Functions
15.4 Hypothesis Tests
15.5 Selecting a Model
Part V: Credibility
Chapter 16: Introduction To Limited Fluctuation Credibility
16.1 Introduction
16.2 Limited Fluctuation Credibility Theory
16.3 Full Credibility
16.4 Partial Credibility
16.5 Problems with the Approach
16.6 Notes and References
16.7 Exercises
Chapter 17: Greatest Accuracy Credibility
17.1 Introduction
17.2 Conditional Distributions and Expectation
17.3 The Bayesian Methodology
17.4 The Credibility Premium
17.5 The Bühlmann Model
17.6 The Bühlmann–Straub Model
17.7 Exact Credibility
17.8 Notes and References
17.9 Exercises
Chapter 18: Empirical Bayes Parameter Estimation
18.1 Introduction
18.2 Nonparametric Estimation
18.3 Semiparametric Estimation
18.4 Notes and References
18.5 Exercises
Part VI: Simulation
Chapter 19: Simulation
19.1 Basics of Simulation
19.2 Simulation for Specific Distributions
19.3 Determining the Sample Size
19.4 Examples of Simulation in Actuarial Modeling
Appendix A: An Inventory of Continuous Distributions
A.1 Introduction
A.2 The Transformed Beta Family
A.3 The 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
References
Index
End User License Agreement
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Chapter 16: Introduction To Limited Fluctuation Credibility
Part V
Credibility
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