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 (?, ?, 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 (?, ?, 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 Buhlmann Model 418
17.6 The Buhlmann–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 (?, ?, 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 (?, ?, 0) Class 506
B.3 The (?, ?, 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
