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 xiii

About the Companion Website xv

Part I Introduction

1 Modeling 3

1.1 The Model-Based Approach 3

1.1.1 The Modeling Process 3

1.1.2 The Modeling Advantage 5

1.2 The Organization of This Book 6

2 Random Variables 9

2.1 Introduction 9

2.2 Key Functions and Four Models 11

2.2.1 Exercises 19

3 Basic Distributional Quantities 21

3.1 Moments 21

3.1.1 Exercises 28

3.2 Percentiles 29

3.2.1 Exercises 31

3.3 Generating Functions and Sums of Random Variables 31

3.3.1 Exercises 33

3.4 Tails of Distributions 33

3.4.1 Classification Based on Moments 33

3.4.2 Comparison Based on Limiting Tail Behavior 34

3.4.3 Classification Based on the Hazard Rate Function 35

3.4.4 Classification Based on the Mean Excess Loss Function 36

3.4.5 Equilibrium Distributions and Tail Behavior 38

3.4.6 Exercises 39

3.5 Measures of Risk 41

3.5.1 Introduction 41

3.5.2 Risk Measures and Coherence 41

3.5.3 Value at Risk 43

3.5.4 Tail Value at Risk 44

3.5.5 Exercises 48

Part II Actuarial Models

4 Characteristics of Actuarial Models 51

4.1 Introduction 51

4.2 The Role of Parameters 51

4.2.1 Parametric and Scale Distributions 52

4.2.2 Parametric Distribution Families 54

4.2.3 Finite Mixture Distributions 54

4.2.4 Data-Dependent Distributions 56

4.2.5 Exercises 59

5 Continuous Models 61

5.1 Introduction 61

5.2 Creating New Distributions 61

5.2.1 Multiplication by a Constant 62

5.2.2 Raising to a Power 62

5.2.3 Exponentiation 64

5.2.4 Mixing 64

5.2.5 Frailty Models 68

5.2.6 Splicing 69

5.2.7 Exercises 70

5.3 Selected Distributions and Their Relationships 74

5.3.1 Introduction 74

5.3.2 Two Parametric Families 74

5.3.3 Limiting Distributions 74

5.3.4 Two Heavy-Tailed Distributions 76

5.3.5 Exercises 77

5.4 The Linear Exponential Family 78

5.4.1 Exercises 80

6 Discrete Distributions 81

6.1 Introduction 81

6.1.1 Exercise 82

6.2 The Poisson Distribution 82

6.3 The Negative Binomial Distribution 85

6.4 The Binomial Distribution 87

6.5 The (𝑎, 𝑏, 0) Class 88

6.5.1 Exercises 91

6.6 Truncation and Modification at Zero 92

6.6.1 Exercises 96

7 Advanced Discrete Distributions 99

7.1 Compound Frequency Distributions 99

7.1.1 Exercises 105

7.2 Further Properties of the Compound Poisson Class 105

7.2.1 Exercises 111

7.3 Mixed-Frequency Distributions 111

7.3.1 The General Mixed-Frequency Distribution 111

7.3.2 Mixed Poisson Distributions 113

7.3.3 Exercises 118

7.4 The Effect of Exposure on Frequency 120

7.5 An Inventory of Discrete Distributions 121

7.5.1 Exercises 122

8 Frequency and Severity with Coverage Modifications 125

8.1 Introduction 125

8.2 Deductibles 126

8.2.1 Exercises 131

8.3 The Loss Elimination Ratio and the Effect of Inflation for Ordinary Deductibles 132

8.3.1 Exercises 133

8.4 Policy Limits 134

8.4.1 Exercises 136

8.5 Coinsurance, Deductibles, and Limits 136

8.5.1 Exercises 138

8.6 The Impact of Deductibles on Claim Frequency 140

8.6.1 Exercises 144

9 Aggregate Loss Models 147

9.1 Introduction 147

9.1.1 Exercises 150

9.2 Model Choices 150

9.2.1 Exercises 151

9.3 The Compound Model for Aggregate Claims 151

9.3.1 Probabilities and Moments 152

9.3.2 Stop-Loss Insurance 157

9.3.3 The Tweedie Distribution 159

9.3.4 Exercises 160

9.4 Analytic Results 167

9.4.1 Exercises 170

9.5 Computing the Aggregate Claims Distribution 171

9.6 The Recursive Method 173

9.6.1 Applications to Compound Frequency Models 175

9.6.2 Underflow/Overflow Problems 177

9.6.3 Numerical Stability 178

9.6.4 Continuous Severity 178

9.6.5 Constructing Arithmetic Distributions 179

9.6.6 Exercises 182

9.7 The Impact of Individual Policy Modifications on Aggregate Payments 186

9.7.1 Exercises 189

9.8 The Individual Risk Model 189

9.8.1 The Model 189

9.8.2 Parametric Approximation 191

9.8.3 Compound Poisson Approximation 193

9.8.4 Exercises 195

Part III Mathematical Statistics

10 Introduction to Mathematical Statistics 201

10.1 Introduction and Four Data Sets 201

10.2 Point Estimation 203

10.2.1 Introduction 203

10.2.2 Measures of Quality 204

10.2.3 Exercises 214

10.3 Interval Estimation 216

10.3.1 Exercises 218

10.4 The Construction of Parametric Estimators 218

10.4.1 The Method of Moments and Percentile Matching 218

10.4.2 Exercises 221

10.5 Tests of Hypotheses 224

10.5.1 Exercise 228

11 Maximum Likelihood Estimation 229

11.1 Introduction 229

11.2 Individual Data 231

11.2.1 Exercises 232

11.3 Grouped Data 235

11.3.1 Exercises 236

11.4 Truncated or Censored Data 236

11.4.1 Exercises 241

11.5 Variance and Interval Estimation for Maximum Likelihood Estimators 242

11.5.1 Exercises 247

11.6 Functions of Asymptotically Normal Estimators 248

11.6.1 Exercises 250

11.7 Nonnormal Confidence Intervals 251

11.7.1 Exercise 253

12 Frequentist Estimation for Discrete Distributions 255

12.1 The Poisson Distribution 255

12.2 The Negative Binomial Distribution 259

12.3 The Binomial Distribution 261

12.4 The (𝑎, 𝑏, 1) Class 264

12.5 Compound Models 268

12.6 The Effect of Exposure on Maximum Likelihood Estimation 269

12.7 Exercises 270

13 Bayesian Estimation 275

13.1 Definitions and Bayes’ Theorem 275

13.2 Inference and Prediction 279

13.2.1 Exercises 285

13.3 Conjugate Prior Distributions and the Linear Exponential Family 290

13.3.1 Exercises 291

13.4 Computational Issues 292

Part IV Construction of Models

14 Construction of Empirical Models 295

14.1 The Empirical Distribution 295

14.2 Empirical Distributions for Grouped Data 300

14.2.1 Exercises 301

14.3 Empirical Estimation with Right Censored Data 304

14.3.1 Exercises 316

14.4 Empirical Estimation of Moments 320

14.4.1 Exercises 326

14.5 Empirical Estimation with Left Truncated Data 327

14.5.1 Exercises 331

14.6 Kernel Density Models 332

14.6.1 Exercises 336

14.7 Approximations for Large Data Sets 337

14.7.1 Introduction 337

14.7.2 Using Individual Data Points 339

14.7.3 Interval-Based Methods 342

14.7.4 Exercises 346

14.8 Maximum Likelihood Estimation of Decrement Probabilities 347

14.8.1 Exercise 349

14.9 Estimation of Transition Intensities 350

15 Model Selection 353

15.1 Introduction 353

15.2 Representations of the Data and Model 354

15.3 Graphical Comparison of the Density and Distribution Functions 355

15.3.1 Exercises 360

15.4 Hypothesis Tests 360

15.4.1 The Kolmogorov–Smirnov Test 360

15.4.2 The Anderson–Darling Test 363

15.4.3 The Chi-Square Goodness-of-Fit Test 363

15.4.4 The Likelihood Ratio Test 367

15.4.5 Exercises 369

15.5 Selecting a Model 371

15.5.1 Introduction 371

15.5.2 Judgment-Based Approaches 372

15.5.3 Score-Based Approaches 373

15.5.4 Exercises 381

Part V Credibility

16 Introduction to Limited Fluctuation Credibility 387

16.1 Introduction 387

16.2 Limited Fluctuation Credibility Theory 389

16.3 Full Credibility 390

16.4 Partial Credibility 393

16.5 Problems with the Approach 397

16.6 Notes and References 397

16.7 Exercises 397

17 Greatest Accuracy Credibility 401

17.1 Introduction 401

17.2 Conditional Distributions and Expectation 404

17.3 The Bayesian Methodology 408

17.4 The Credibility Premium 415

17.5 The Bühlmann Model 418

17.6 The Bühlmann–Straub Model 422

17.7 Exact Credibility 427

17.8 Notes and References 431

17.9 Exercises 432

18 Empirical Bayes Parameter Estimation 445

18.1 Introduction 445

18.2 Nonparametric Estimation 448

18.3 Semiparametric Estimation 459

18.4 Notes and References 460

18.5 Exercises 460

Part VI Simulation

19 Simulation 467

19.1 Basics of Simulation 467

19.1.1 The Simulation Approach 468

19.1.2 Exercises 472

19.2 Simulation for Specific Distributions 472

19.2.1 Discrete Mixtures 472

19.2.2 Time or Age of Death from a Life Table 473

19.2.3 Simulating from the (𝑎, 𝑏, 0) Class 474

19.2.4 Normal and Lognormal Distributions 476

19.2.5 Exercises 477

19.3 Determining the Sample Size 477

19.3.1 Exercises 479

19.4 Examples of Simulation in Actuarial Modeling 480

19.4.1 Aggregate Loss Calculations 480

19.4.2 Examples of Lack of Independence 480

19.4.3 Simulation Analysis of the Two Examples 481

19.4.4 The Use of Simulation to Determine Risk Measures 484

19.4.5 Statistical Analyses 484

19.4.6 Exercises 486

A An Inventory of Continuous Distributions 489

A.1 Introduction 489

A.2 The Transformed Beta Family 493

A.2.1 The Four-Parameter Distribution 493

A.2.2 Three-Parameter Distributions 493

A.2.3 Two-Parameter Distributions 494

A.3 The Transformed Gamma Family 496

A.3.1 Three-Parameter Distributions 496

A.3.2 Two-Parameter Distributions 497

A.3.3 One-Parameter Distributions 499

A.4 Distributions for Large Losses 499

A.4.1 Extreme Value Distributions 499

A.4.2 Generalized Pareto Distributions 500

A.5 Other Distributions 501

A.6 Distributions with Finite Support 502

B An Inventory of Discrete Distributions 505

B.1 Introduction 505

B.2 The (𝑎, 𝑏, 0) Class 506

B.3 The (𝑎, 𝑏, 1) Class 507

B.3.1 The Zero-Truncated Subclass 507

B.3.2 The Zero-Modified Subclass 509

B.4 The Compound Class 509

B.4.1 Some Compound Distributions 510

B.5 A Hierarchy of Discrete Distributions 511

C Frequency and Severity Relationships 513

D The Recursive Formula 515

E Discretization of the Severity Distribution 517

E.1 The Method of Rounding 517

E.2 Mean Preserving 518

E.3 Undiscretization of a Discretized Distribution 518

References 521

Index 529

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