---
title: "Loss Models: From Data to Decisions, Fifth Edition"
tags:
  - authorship/other
  - exclude-from-word-count
  - topic/risk
  - type/media/book
authors:
  - Gordon E. Willmot
  - Harry H. Panjer
  - Stuart A. Klugman
edition: Fifth
publisher: John	Wiley & Sons, Inc.
series: Wiley Series in Probability and Statistics
subtitle: From Data to Decisions
type: book
year: 2019
---
# Loss Models: From Data to Decisions, Fifth Edition

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consists only of content from the text.
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## 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

## References

## Index
