title, tags, author, edition, publisher, series, subtitle, type, year
title
tags
author
edition
publisher
series
subtitle
type
year
Loss Models: From Data to Decisions, Fifth Edition
authorship/other
exclude-from-word-count
topic/risk
type/media/book
Gordon E. Willmot & Harry H. Panjer & 3
Fifth
John Wiley & Sons
Wiley Series in Probability and Statistics
From Data to Decisions
book
2019
Loss Models: From Data to Decisions, Fifth Edition
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This note, with the exception of comments like this one
(reserved for notes on transcription)
consists only of content from the text.
For commentary see the companion loss-models.
%%
Part I Introduction
1. Modeling
1.1 The Model-Based Approach
1.1.1 The Modeling Process
The model-based approach
should be considered in the context of the objectives of any given problem.
Many problems in actuarial science
involve the building of a mathematical model
that can be used to forecast or predict insurance costs in the future.
1.1.2 The Modeling Advantage
1.2 The Organization of This Book
2. Random Variables
2.1 Introduction
2.2 Key Functions and Four Models
2.2.1 Exercises
3. Basic Distributional Quantities
3.1 Moments
3.1.1 Exercises
3.2 Percentiles
3.2.1 Exercises
3.3 Generating Functions and Sums of Random Variables
3.3.1 Exercises
3.4 Tails of Distributions
3.4.1 Classification Based on Moments
3.4.2 Comparison Based on Limiting Tail
3.4.3 Classification Based on the Hazard Rate Function
3.4.4 Classification Based on the Mean Excess Loss Function The mean excess
3.4.5 Equilibrium Distributions and Tail Behavior
3.4.6 Exercises
3.5 Measures of Risk
3.5.1 Introduction
3.5.2 Risk Measures and Coherence
3.5.3 Value at Risk
3.5.4 Tail Value of Risk
3.5.5 Exercises
Part II Actuarial Models
4. Characteristics Of Actuarial Models
4.1 Introduction
4.2 The Role of Parameters
4.2.1 Parametric and Scale Distributions
4.2.2 Parametric Distribution Families
4.2.3 Finite Mixture Distributions
4.2.4 Data-Dependent Distributions
4.2.5 Exercises
5. Continuous Models
5.1 Introduction
5.2 Creating New Distributions
5.2.1 Multiplication by a Constant
5.2.2 Raising to a Power
5.2.3 Exponentiation
5.2.4 Mixing
5.2.5 Frailty Models
5.2.6 Splicing
5.2.7 Exercises
5.3 Selected Distributions and Their Relationships
5.3.1 Introduction
5.3.2 Two Parametric Families
5.3.3 Limiting Distributions
5.3.4 Two Heavy-Tailed Distributions
5.3.5 Exercises
5.4 The Linear Exponential Family
5.4.1 Exercises
6. Discrete Distributions
6.1 Introduction
6.1.1 Exercise
6.2 The Poisson Distribution
6.3 The Negative Binomial Distribution
6.4 The Binomial Distribution
6.5 The (a,b) Class
6.5.1 Exercises
6.6 Truncation and Modification at zero
6.6.1 Exercises
7. Advanced Discrete Distributions
7.1 Compound Frequency Distributions
7.1.1 Exercises
7.2 Further Properties of the Compound Poisson Class
7.2.1 Exercises
7.3 Mixed-Frequency Distributions
7.3.1 The General Mixed-Frequency Distribution
7.3.2 Mixed Poisson Distributions
7.3.3 Exercises
7.4 The Effect of Exposure on Frequency
7.5 An Inventory of Discrete Distributions
7.5.1 Exercises
8. Frequency And Severity With Coverage Modifications
8.1 Introduction
8.2 Deductibles
8.2.1 Exercises
8.3 The Loss Elimination Ratio and the Effect of Inflation for Ordinary Deductibles
8.3.1 Exercise
8.4 Policy Limits
8.4.1 Exercises
8.5 Coinsurance, Deductibles, and Limits
8.5.1 Exercises
8.6 The Impact of Deductibles on Claim Frequency
8.6.1 Exercises
9. Aggregate Loss Models
9.1 Introduction
9.1.1 Exercises
9.2 Model Choices
9.2.1 Exercises
9.3 The Compound Model for Aggregate Claims
9.3.1 Probabilities and Moments
9.3.2 Stop-Loss Insurance
9.3.3 The Tweedle Distribution
9.3.4 Exercises
9.4 Analytic Results
9.4.1 Exercises
9.5 Computing the Aggregate Claims Distribution
9.6 The Recursive Method
9.6.1 Applications to Compound Frequency Models
9.6.2 Underflow/Overflow Problems
9.6.3 Numerical Stability
9.6.4 Continuous Severity
9.6.5 Constructing Arithmetic Distributions
9.6.6 Exercises
9.7 The Impact of Individual Policy Modifications on Aggregate Payments
9.7.1 Exercises
9.8 The individual Risk Model
9.8.1 The Model
9.8.2 Parametric Approximation
9.8.3 Compound Poisson Approximation
9.8.4 Exercises
Part III Mathematical Statistics
10. Introduction To Mathematical Statistics
10.1 Introduction and Four Data Sets
10.2 Point Estimation
10.2.1 Introduction
10.2.2 Measures of Quality
10.2.3 Exercises
10.3 Interval Estimation
10.3.1 Exercises
10.4 The Construction of Parametric Estimators
10.4.1 The Method of Moments and Percentile Matching
10.4.2 Exercises
10.5 Tests of Hypotheses
10.5.1 Exercise
11. Maximum Likelihood Estimation
11.1 Introduction
11.2 Individual Data
11.2.1 Exercises
11.3 Grouped Data
11.3.1 Exercises
11.4 Truncated or Censored Data
11.4.1 Exercises
11.5 Variance and Interval Estimation for Maximum Likelihood Estimators
11.5.1 Exercises
11.6 Functions of Asymptotically Normal Estimators
11.6.1 Exercises
11.7 Nonnormal Confidence Intervals
11.7.1 Exercise
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) Class
12.5 Compound Models
12.6 The Effect of Exposure on Maximum Likelihood Estimation
12.7 Exercises
13. Bayesian Estimation
13.1 Definitions and Bayes' Theorem
13.2 Inference and Prediction
13.2.1 Exercises
13.3 Conjugate Prior Distributions and the Linear Exponential Family
13.3.1 Exercises
13.4 Computational Issues
Part IV Construction Of Models
14. Construction Of Empirical Models
14.1 The Empirical Distribution
14.2 Empirical Distributions for Grouped Data
14.2.1 Exercises
14.3 Empirical Estimation with Right Censored Data
14.3.1 Exercises
14.4 Empirical Estimation of Moments
14.4.1 Exercises
14.5 Empirical Estimation with Left Truncated Data
14.5.1 Exercises
14.6 Kernel Density Models
14.6.1 Exercises
14.7 Approximations for large Data Sets
14.7.1 Introduction
14.7.2 Using Individual Data Points
14.7.3 Interval-Based Methods
14.7.4 Exercises
14.8 Maximum Likelihood Estimation of Decrement Probabilities
14.8.1 Exercise
14.9 Estimation of Transition Intensities
15. Model Selection
15.1 Introduction
15.2 Representations of the Data and Model
15.3 Graphical Comparison of the Density and Distribution Functions
15.3.1 Exercises
15.4 Hypothesis Tests
15.4.1 The Kolmogorov--Smirnov Test
15.4.2 The Anderson--Darling Test
15.4.3 The Chi-Square Goodness-of-Fit Test
15.4.4 The Likelihood Ratio Test
15.4.5 Exercises
15.5 Selecting a Model
15.5.1 Introduction
15.2.2 Judgement-Based Approaches
15.5.3 Score-Based Approaches
15.5.4 Exercises
Part V Credibility
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
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
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
19. Simulation
19.1 Basics of Simulation
19.1.1 The Simulation Approach
19.1.2 Exercises
19.2 Simulation for Specific Distributions
19.2.1 Discrete Mixtures
19.2.2 Time or Age of Death from a Life Table
19.2.3 Simulating from the (a,b) Class
19.2.4 Normal and Lognormal Distributions
19.2.5 Exercises
19.3 Determining the Sample Size
19.3.1 Exercises
19.4 Examples of Simulation in Actuarial Modeling
19.4.1 Aggregate Loss Calculations
19.4.2 Examples of Lack of Independence
19.4.3 Simulation Analysis of the Two Examples
19.4.4 The Use of Simulation to Determine Risk Measures
19.4.5 Statistical Analyses
19.4.6 Exercises
Appendix A An Inventory Of Continuous Distributions
A.1 Introduction
A.2 The Transformed Beta Family
A.2.1 The Four-Parameter Distribution
A.2.2 Three-Parameter Distributions
A.2.3 Two-Parameter Distributions
A.3 The Transformed Gamma Family
A.3.1 Three-Parameter Distributions
A.3.2 Two-Parameter Distributions
A.3.3 One-Parameter Distributions
A.4 Distributions For Large Losses
A.4.1 Extreme Value Distributions
A.4.2 Generalized Pareto Distributions
A.4.5 Other Distributions
A.4.6 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.3.1 The Zero-Truncated Subclass
B.3.2 The Zero-Modified Subclass
B.4 The Compound Class
B.4.1 Some Compound Distributions
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
