STUART A. KLUGMAN, PHD, FSA, CERA, is Staff Fellow (Education) at the Society of Actuaries (SOA) and Principal Financial Group Distinguished Professor Emeritus of Actuarial Science at Drake University. He has served as SOA vice president. HARRY H. PANJER, PHD, FSA, FCIA, CERA, HonFIA, is Distinguished Professor Emeritus in the Department of Statistics and Actuarial Science at the University of Waterloo, Canada. He has served as CIA president and as SOA president. GORDON E. WILLMOT, PHD, FSA, FCIA, is Munich Re Chair in Insurance and Professor in the Department of Statistics and Actuarial Science at the University of Waterloo, Canada.

• Preface xiiiAbout the Companion Website xvPart I Introduction1 Modeling 31.1 The Model-Based Approach 31.1.1 The Modeling Process 31.1.2 The Modeling Advantage 51.2 The Organization of This Book 62 Random Variables 92.1 Introduction 92.2 Key Functions and Four Models 112.2.1 Exercises 193 Basic Distributional Quantities 213.1 Moments 213.1.1 Exercises 283.2 Percentiles 293.2.1 Exercises 313.3 Generating Functions and Sums of Random Variables 313.3.1 Exercises 333.4 Tails of Distributions 333.4.1 Classification Based on Moments 333.4.2 Comparison Based on Limiting Tail Behavior 343.4.3 Classification Based on the Hazard Rate Function 353.4.4 Classification Based on the Mean Excess Loss Function 363.4.5 Equilibrium Distributions and Tail Behavior 383.4.6 Exercises 393.5 Measures of Risk 413.5.1 Introduction 413.5.2 Risk Measures and Coherence 413.5.3 Value at Risk 433.5.4 Tail Value at Risk 443.5.5 Exercises 48Part II Actuarial Models4 Characteristics of Actuarial Models 514.1 Introduction 514.2 The Role of Parameters 514.2.1 Parametric and Scale Distributions 524.2.2 Parametric Distribution Families 544.2.3 Finite Mixture Distributions 544.2.4 Data-Dependent Distributions 564.2.5 Exercises 595 Continuous Models 615.1 Introduction 615.2 Creating New Distributions 615.2.1 Multiplication by a Constant 625.2.2 Raising to a Power 625.2.3 Exponentiation 645.2.4 Mixing 645.2.5 Frailty Models 685.2.6 Splicing 695.2.7 Exercises 705.3 Selected Distributions and Their Relationships 745.3.1 Introduction 745.3.2 Two Parametric Families 745.3.3 Limiting Distributions 745.3.4 Two Heavy-Tailed Distributions 765.3.5 Exercises 775.4 The Linear Exponential Family 785.4.1 Exercises 806 Discrete Distributions 816.1 Introduction 816.1.1 Exercise 826.2 The Poisson Distribution 826.3 The Negative Binomial Distribution 856.4 The Binomial Distribution 876.5 The (𝑎, 𝑏, 0) Class 886.5.1 Exercises 916.6 Truncation and Modification at Zero 926.6.1 Exercises 967 Advanced Discrete Distributions 997.1 Compound Frequency Distributions 997.1.1 Exercises 1057.2 Further Properties of the Compound Poisson Class 1057.2.1 Exercises 1117.3 Mixed-Frequency Distributions 1117.3.1 The General Mixed-Frequency Distribution 1117.3.2 Mixed Poisson Distributions 1137.3.3 Exercises 1187.4 The Effect of Exposure on Frequency 1207.5 An Inventory of Discrete Distributions 1217.5.1 Exercises 1228 Frequency and Severity with Coverage Modifications 1258.1 Introduction 1258.2 Deductibles 1268.2.1 Exercises 1318.3 The Loss Elimination Ratio and the Effect of Inflation for Ordinary Deductibles 1328.3.1 Exercises 1338.4 Policy Limits 1348.4.1 Exercises 1368.5 Coinsurance, Deductibles, and Limits 1368.5.1 Exercises 1388.6 The Impact of Deductibles on Claim Frequency 1408.6.1 Exercises 1449 Aggregate Loss Models 1479.1 Introduction 1479.1.1 Exercises 1509.2 Model Choices 1509.2.1 Exercises 1519.3 The Compound Model for Aggregate Claims 1519.3.1 Probabilities and Moments 1529.3.2 Stop-Loss Insurance 1579.3.3 The Tweedie Distribution 1599.3.4 Exercises 1609.4 Analytic Results 1679.4.1 Exercises 1709.5 Computing the Aggregate Claims Distribution 1719.6 The Recursive Method 1739.6.1 Applications to Compound Frequency Models 1759.6.2 Underflow/Overflow Problems 1779.6.3 Numerical Stability 1789.6.4 Continuous Severity 1789.6.5 Constructing Arithmetic Distributions 1799.6.6 Exercises 1829.7 The Impact of Individual Policy Modifications on Aggregate Payments 1869.7.1 Exercises 1899.8 The Individual Risk Model 1899.8.1 The Model 1899.8.2 Parametric Approximation 1919.8.3 Compound Poisson Approximation 1939.8.4 Exercises 195Part III Mathematical Statistics10 Introduction to Mathematical Statistics 20110.1 Introduction and Four Data Sets 20110.2 Point Estimation 20310.2.1 Introduction 20310.2.2 Measures of Quality 20410.2.3 Exercises 21410.3 Interval Estimation 21610.3.1 Exercises 21810.4 The Construction of Parametric Estimators 21810.4.1 The Method of Moments and Percentile Matching 21810.4.2 Exercises 22110.5 Tests of Hypotheses 22410.5.1 Exercise 22811 Maximum Likelihood Estimation 22911.1 Introduction 22911.2 Individual Data 23111.2.1 Exercises 23211.3 Grouped Data 23511.3.1 Exercises 23611.4 Truncated or Censored Data 23611.4.1 Exercises 24111.5 Variance and Interval Estimation for Maximum Likelihood Estimators 24211.5.1 Exercises 24711.6 Functions of Asymptotically Normal Estimators 24811.6.1 Exercises 25011.7 Nonnormal Confidence Intervals 25111.7.1 Exercise 25312 Frequentist Estimation for Discrete Distributions 25512.1 The Poisson Distribution 25512.2 The Negative Binomial Distribution 25912.3 The Binomial Distribution 26112.4 The (𝑎, 𝑏, 1) Class 26412.5 Compound Models 26812.6 The Effect of Exposure on Maximum Likelihood Estimation 26912.7 Exercises 27013 Bayesian Estimation 27513.1 Definitions and Bayes’ Theorem 27513.2 Inference and Prediction 27913.2.1 Exercises 28513.3 Conjugate Prior Distributions and the Linear Exponential Family 29013.3.1 Exercises 29113.4 Computational Issues 292Part IV Construction of Models14 Construction of Empirical Models 29514.1 The Empirical Distribution 29514.2 Empirical Distributions for Grouped Data 30014.2.1 Exercises 30114.3 Empirical Estimation with Right Censored Data 30414.3.1 Exercises 31614.4 Empirical Estimation of Moments 32014.4.1 Exercises 32614.5 Empirical Estimation with Left Truncated Data 32714.5.1 Exercises 33114.6 Kernel Density Models 33214.6.1 Exercises 33614.7 Approximations for Large Data Sets 33714.7.1 Introduction 33714.7.2 Using Individual Data Points 33914.7.3 Interval-Based Methods 34214.7.4 Exercises 34614.8 Maximum Likelihood Estimation of Decrement Probabilities 34714.8.1 Exercise 34914.9 Estimation of Transition Intensities 35015 Model Selection 35315.1 Introduction 35315.2 Representations of the Data and Model 35415.3 Graphical Comparison of the Density and Distribution Functions 35515.3.1 Exercises 36015.4 Hypothesis Tests 36015.4.1 The Kolmogorov–Smirnov Test 36015.4.2 The Anderson–Darling Test 36315.4.3 The Chi-Square Goodness-of-Fit Test 36315.4.4 The Likelihood Ratio Test 36715.4.5 Exercises 36915.5 Selecting a Model 37115.5.1 Introduction 37115.5.2 Judgment-Based Approaches 37215.5.3 Score-Based Approaches 37315.5.4 Exercises 381Part V Credibility16 Introduction to Limited Fluctuation Credibility 38716.1 Introduction 38716.2 Limited Fluctuation Credibility Theory 38916.3 Full Credibility 39016.4 Partial Credibility 39316.5 Problems with the Approach 39716.6 Notes and References 39716.7 Exercises 39717 Greatest Accuracy Credibility 40117.1 Introduction 40117.2 Conditional Distributions and Expectation 40417.3 The Bayesian Methodology 40817.4 The Credibility Premium 41517.5 The Bühlmann Model 41817.6 The Bühlmann–Straub Model 42217.7 Exact Credibility 42717.8 Notes and References 43117.9 Exercises 43218 Empirical Bayes Parameter Estimation 44518.1 Introduction 44518.2 Nonparametric Estimation 44818.3 Semiparametric Estimation 45918.4 Notes and References 46018.5 Exercises 460Part VI Simulation19 Simulation 46719.1 Basics of Simulation 46719.1.1 The Simulation Approach 46819.1.2 Exercises 47219.2 Simulation for Specific Distributions 47219.2.1 Discrete Mixtures 47219.2.2 Time or Age of Death from a Life Table 47319.2.3 Simulating from the (𝑎, 𝑏, 0) Class 47419.2.4 Normal and Lognormal Distributions 47619.2.5 Exercises 47719.3 Determining the Sample Size 47719.3.1 Exercises 47919.4 Examples of Simulation in Actuarial Modeling 48019.4.1 Aggregate Loss Calculations 48019.4.2 Examples of Lack of Independence 48019.4.3 Simulation Analysis of the Two Examples 48119.4.4 The Use of Simulation to Determine Risk Measures 48419.4.5 Statistical Analyses 48419.4.6 Exercises 486A An Inventory of Continuous Distributions 489A.1 Introduction 489A.2 The Transformed Beta Family 493A.2.1 The Four-Parameter Distribution 493A.2.2 Three-Parameter Distributions 493A.2.3 Two-Parameter Distributions 494A.3 The Transformed Gamma Family 496A.3.1 Three-Parameter Distributions 496A.3.2 Two-Parameter Distributions 497A.3.3 One-Parameter Distributions 499A.4 Distributions for Large Losses 499A.4.1 Extreme Value Distributions 499A.4.2 Generalized Pareto Distributions 500A.5 Other Distributions 501A.6 Distributions with Finite Support 502B An Inventory of Discrete Distributions 505B.1 Introduction 505B.2 The (𝑎, 𝑏, 0) Class 506B.3 The (𝑎, 𝑏, 1) Class 507B.3.1 The Zero-Truncated Subclass 507B.3.2 The Zero-Modified Subclass 509B.4 The Compound Class 509B.4.1 Some Compound Distributions 510B.5 A Hierarchy of Discrete Distributions 511C Frequency and Severity Relationships 513D The Recursive Formula 515E Discretization of the Severity Distribution 517E.1 The Method of Rounding 517E.2 Mean Preserving 518E.3 Undiscretization of a Discretized Distribution 518References 521Index 529