Table of contents for Loss models : from data to decisions / Stuart A. Klugman, Harry H. Panjer, Gordon E. Willmot.

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CONTENTS
Preface xix
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 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 20
3 Basic distributional quantities 21
3.1 Moments 21
3.1.1 Exercises 28
3.2 Quantiles 29
3.2.1 Exercises 30
3.3 Generating functions and sums of random variables 31
3.3.1 Exercises 35
3.4 Tails of distributions 35
3.4.1 Classification based on moments 36
3.4.2 Comparison based on limiting tail behavior 37
3.4.3 Classification based on the hazard rate function 37
3.4.4 Classification based on the mean excess loss function 39
3.4.5 Equilibrium distributions and tail behavior 40
3.4.6 Exercises 41
3.5 Measures of Risk 43
3.5.1 Introduction 43
3.5.2 Risk measures and coherence 43
3.5.3 Value-at-Risk 45
3.5.4 Tail-Value-at-Risk 47
3.5.5 Exercises 51
PART II ACTUARIAL MODELS
4 Characteristics of Actuarial Models 55
4.1 Introduction 55
4.2 The role of parameters 55
4.2.1 Parametric and scale distributions 56
4.2.2 Parametric distribution families 58
4.2.3 Finite mixture distributions 58
4.2.4 Data-dependent distributions 60
4.2.5 Exercises 62
5 Continuousmodels 65
5.1 Introduction 65
5.2 Creating new distributions 65
5.2.1 Multiplication by a constant 66
5.2.2 Raising to a power 66
5.2.3 Exponentiation 68
5.2.4 Mixing 68
5.2.5 Frailty models 72
5.2.6 Splicing 73
5.2.7 Exercises 75
5.3 Selected distributions and their relationships 78
5.3.1 Introduction 78
5.3.2 Two parametric families 79
5.3.3 Limiting distributions 80
5.3.4 Exercises 81
5.4 The linear exponential family 81
5.4.1 Exercises 84
5.5 TVaR for continuous distributions 84
5.5.1 Continuous elliptical distributions 85
5.5.2 TVaR for the linear exponential family 87
5.5.3 Exercise 89
5.6 Extreme value distributions 89
5.6.1 Introduction 89
5.6.2 Distribution of the maximum 91
5.6.3 Stability of the maximum of the extreme value
distribution 95
5.6.4 The Fisher-Tippett theorem 96
5.6.5 Maximum domain of attraction 98
5.6.6 Generalized Pareto distributions 101
5.6.7 Stability of excesses of the generalized Pareto 103
5.6.8 Limiting distributions of excesses 104
5.6.9 TVaR for extreme value distributions 105
5.6.10 Further reading 106
5.6.11 Exercises 106
6 Discrete distributions and processes 109
6.1 Introduction 109
6.1.1 Exercise 110
6.2 The Poisson distribution 110
6.3 The negative binomial distribution 113
6.4 The binomial distribution 115
6.5 The (a, b, 0) class 117
6.5.1 Exercises 120
6.6 Counting processes 120
6.6.1 Introduction and definitions 120
6.6.2 Poisson processes 123
6.6.3 Processes with contagion 125
6.6.4 Other processes 128
6.6.5 Exercises 129
6.7 Truncation and modification at zero 130
6.7.1 Exercises 135
6.8 Compound frequency models 135
6.8.1 Exercises 142
6.9 Further properties of the compound Poisson class 142
6.9.1 Exercises 147
6.10 Mixed Poisson distributions 147
6.10.1 General mixed frequency distribution 147
6.10.2 Mixed Poisson distributions 149
6.10.3 Exercises 154
6.11 Mixed Poisson processes 156
6.11.1 Exercises 160
6.12 Effect of exposure on frequency 162
6.13 An inventory of discrete distributions 163
6.13.1 Exercises 164
6.14 TVaR for discrete distributions 166
6.14.1 TVaR for the discrete linear exponential family 166
6.14.2 Exercises 169
7 Multivariatemodels 171
7.1 Introduction 171
7.2 Sklar?s theorem and copulas 172
7.3 Measures of dependency 174
7.3.1 Spearman?s rho 174
7.3.2 Kendall?s tau 175
7.4 Tail dependence 175
7.5 Archimedean copulas 176
7.5.1 Exercise 182
7.6 Elliptical copulas 182
7.6.1 Exercises 184
7.7 Extreme value copulas 185
7.7.1 Exercises 188
7.8 Archimax copulas 188
8 Frequency and severity with coverage modifications 191
8.1 Introduction 191
8.2 Deductibles 191
8.2.1 Exercises 196
8.3 The loss elimination ratio and the effect of inflation for ordinary
deductibles 197
8.3.1 Exercises 199
8.4 Policy limits 199
8.4.1 Exercises 201
8.5 Coinsurance, deductibles, and limits 202
8.5.1 Exercises 204
8.6 The impact of deductibles on claim frequency 205
8.6.1 Exercises 209
9 Aggregate loss models 211
9.1 Introduction 211
9.1.1 Exercises 214
9.2 Model choices 215
9.2.1 Exercises 215
9.3 The compound model for aggregate claims 216
9.3.1 Exercises 223
9.4 Analytic results 230
9.4.1 Exercises 235
9.5 Computing the aggregate claims distribution 238
9.6 The recursive method 240
9.6.1 Applications to compound frequency models 241
9.6.2 Underflow/overflow problems 243
9.6.3 Numerical stability 243
9.6.4 Continuous severity 244
9.6.5 Constructing arithmetic distributions 244
9.6.6 Exercises 247
9.7 The impact of individual policy modifications on aggregate
payments 251
9.7.1 Exercises 254
9.8 Inversion methods 255
9.8.1 Fast Fourier transform 255
9.8.2 Direct numerical inversion 258
9.8.3 Exercise 260
9.9 Calculations with approximate distributions 260
9.9.1 Arithmetic distributions 260
9.9.2 Empirical distributions 263
9.9.3 Piecewise linear cdf 264
9.9.4 Exercises 265
9.10 Comparison of methods 266
9.11 The individual risk model 267
9.11.1 The model 267
9.11.2 Parametric approximation 269
9.11.3 Compound Poisson approximation 271
9.11.4 Exercises 273
9.12 TVaR for aggregate losses 275
9.12.1 TVaR for discrete aggregate loss distributions 276
9.12.2 Aggregate TVaR for some frequency distributions 277
9.12.3 Aggregate TVaR for some severity distributions 278
9.12.4 Summary 282
9.12.5 Exercises 282
10 Discrete-time ruin models 285
10.1 Introduction 285
10.2 Process models for insurance 286
10.2.1 Processes 286
10.2.2 An insurance model 287
10.2.3 Ruin 289
10.3 Discrete, finite-time ruin probabilities 290
10.3.1 The discrete-time process 290
10.3.2 Evaluating the probability of ruin 291
10.3.3 Exercises 293
11 Continuous-time ruin models 295
11.1 Introduction 295
11.1.1 The Poisson process 295
11.1.2 The continuous-time problem 296
11.2 The adjustment coefficient and Lundberg?s inequality 297
11.2.1 The adjustment coefficient 297
11.2.2 Lundberg?s inequality 301
11.2.3 Exercises 303
11.3 An integrodifferential equation 305
11.3.1 Exercises 309
11.4 The maximum aggregate loss 310
11.4.1 Exercises 312
11.5 Cramer?s asymptotic ruin formula and Tijms? approximation 314
11.5.1 Exercises 319
11.6 The Brownian motion risk process 321
11.7 Brownian motion and the probability of ruin 325
PART III CONSTRUCTION OF EMPIRICAL MODELS
12 Review of mathematical statistics 333
12.1 Introduction 333
12.2 Point estimation 334
12.2.1 Introduction 334
12.2.2 Measures of quality 335
12.2.3 Exercises 341
12.3 Interval estimation 342
12.3.1 Exercises 344
12.4 Tests of hypotheses 345
12.4.1 Exercise 349
13 Estimation for complete data 351
13.1 Introduction 351
13.2 The empirical distribution for complete, individual data 355
13.2.1 Exercises 359
13.3 Empirical distributions for grouped data 359
13.3.1 Exercises 361
14 Estimation for modified data 365
14.1 Point estimation 365
14.1.1 Exercises 371
14.2 Means, variances, and interval estimation 373
14.2.1 Exercises 382
14.3 Kernel density models 384
14.3.1 Exercises 387
14.4 Approximations for large data sets 389
14.4.1 Introduction 389
14.4.2 Kaplan?Meier type approximations 390
14.4.3 Exercises 393
PART IV PARAMETRIC STATISTICAL METHODS
15 Parameter estimation 399
15.1 Method of moments and percentile matching 399
15.1.1 Exercises 403
15.2 Maximum likelihood estimation 405
15.2.1 Introduction 405
15.2.2 Complete, individual data 407
15.2.3 Complete, grouped data 408
15.2.4 Truncated or censored data 409
15.2.5 Exercises 412
15.3 Variance and interval estimation 418
15.3.1 Exercises 424
15.4 Non-normal confidence intervals 426
15.4.1 Exercise 429
15.5 Bayesian estimation 429
15.5.1 Definitions and Bayes? theorem 429
15.5.2 Inference and prediction 432
15.5.3 Conjugate prior distributions and the linear exponential
family 439
15.5.4 Computational issues 440
15.5.5 Exercises 441
15.6 Estimation for discrete distributions 448
15.6.1 Poisson 448
15.6.2 Negative binomial 451
15.6.3 Binomial 453
15.6.4 The (a, b, 1) class 456
15.6.5 Compound models 460
15.6.6 Effect of exposure on maximum likelihood estimation 462
15.6.7 Exercises 463
16 Model selection 467
16.1 Introduction 467
16.2 Representations of the data and model 468
16.3 Graphical comparison of the density and distribution functions 469
16.3.1 Exercises 473
16.4 Hypothesis tests 473
16.4.1 Kolmogorov?Smirnov test 475
16.4.2 Anderson?Darling test 477
16.4.3 Chi-square goodness-of-fit test 478
16.4.4 Likelihood ratio test 482
16.4.5 Exercises 484
16.5 Selecting a model 486
16.5.1 Introduction 486
16.5.2 Judgment-based approaches 487
16.5.3 Score-based approaches 487
16.5.4 Exercises 495
17 Estimation and model selection for more complex models 501
17.1 Extreme value models 501
17.1.1 Introduction 501
17.1.2 Parameter estimation 502
17.2 Copula models 511
17.2.1 Introduction 512
17.2.2 Maximum likelihood estimation 512
17.2.3 Semiparametric estimation of the copula 514
17.2.4 The role of deductibles 515
17.2.5 Goodness-of-fit testing 517
17.2.6 An example 518
17.2.7 Exercise 519
17.3 Models with covariates 520
17.3.1 Introduction 520
17.3.2 Proportional hazards models 521
17.3.3 The generalized linear and accelerated failure time
models 527
17.3.4 Exercises 530
18 Five examples 533
18.1 Introduction 533
18.2 Time to death 533
18.2.1 The data 533
18.2.2 Some calculations 534
18.2.3 Exercise 537
18.3 Time from incidence to report 537
18.3.1 The problem and some data 537
18.3.2 Analysis 537
18.4 Payment amount 539
18.4.1 The data 539
18.4.2 The first model 540
18.4.3 The second model 542
18.5 An aggregate loss example 543
18.6 Another aggregate loss example 547
18.6.1 Distribution for a single policy 547
18.6.2 One hundred policies?excess of loss 548
18.6.3 One hundred policies?aggregate stop-loss 549
18.6.4 Numerical convolutions 550
18.7 Comprehensive exercises 552
PART V ADJUSTED ESTIMATES
19 Interpolation and smoothing 559
19.1 Introduction 559
19.2 Polynomial interpolation and smoothing 561
19.2.1 Exercises 565
19.3 Cubic spline interpolation 565
19.3.1 Construction of cubic splines 566
19.3.2 Exercises 574
19.4 Approximating functions with splines 575
19.4.1 Exercise 578
19.5 Extrapolating with splines 578
19.5.1 Exercise 579
19.6 Smoothing splines 579
19.6.1 Exercise 586
20 Credibility 589
20.1 Introduction 589
20.2 Limited fluctuation credibility theory 591
20.2.1 Full credibility 592
20.2.2 Partial credibility 595
20.2.3 Problems with the approach 599
20.2.4 Notes and References 599
20.2.5 Exercises 599
20.3 Greatest accuracy credibility theory 602
20.3.1 Introduction 602
20.3.2 Conditional distributions and expectation 604
20.3.3 The Bayesian methodology 608
20.3.4 The credibility premium 616
20.3.5 The Buhlmann model 619
20.3.6 The Buhlmann?Straub model 623
20.3.7 Exact credibility 629
20.3.8 Linear versus Bayesian versus no credibility 633
20.3.9 Notes and References 640
20.3.10 Exercises 641
20.4 Empirical Bayes parameter estimation 654
20.4.1 Nonparametric estimation 656
20.4.2 Semiparametric estimation 668
20.4.3 Parametric estimation 669
20.4.4 Notes and References 674
20.4.5 Exercises 674
PART VI SIMULATION
21 Simulation 681
21.1 Basics of simulation 681
21.1.1 The simulation approach 682
21.1.2 Exercises 687
21.2 Examples of simulation in actuarial modeling 688
21.2.1 Aggregate loss calculations 688
21.2.2 Examples of lack of independence or identical
distributions 688
21.2.3 Simulation analysis of the two examples 689
21.2.4 Simulating copulas 692
21.2.5 Statistical analyses 694
21.2.6 Exercises 696
21.3 Examples of simulation in finance 698
21.3.1 Investment guarantees 699
21.3.2 Option valuation 700
21.3.3 Exercise 702
Appendix A: An inventory of continuous distributions 703
A.1 Introduction 703
A.2 Transformed beta family 706
A.2.1 Four-parameter distribution 707
A.2.2 Three-parameter distributions 707
A.2.3 Two-parameter distributions 708
A.3 Transformed gamma family 711
A.3.1 Three-parameter distributions 711
A.3.2 Two-parameter distributions 711
A.3.3 One-parameter distributions 713
A.4 Distributions for large losses 714
A.4.1 Extreme value distributions 714
A.4.2 Generalized Pareto distributions 714
A.5 Other distributions 715
A.6 Distributions with finite support 716
Appendix B: An inventory of discrete distributions 719
B.1 Introduction 719
B.2 The (a, b, 0) class 720
B.3 The (a, b, 1) class 721
B.3.1 The zero-truncated subclass 721
B.3.2 The zero-modified subclass 723
B.4 The compound class 723
B.4.1 Some compound distributions 724
B.5 A hierarchy of discrete distributions 725
Appendix C: Frequency and severity relationships 727
Appendix D: The recursive formula 729
Appendix E: Discretization of the severity distribution 731
E.1 The method of rounding 731
E.2 Mean preserving 732
E.3 Undiscretization of a discretized distribution 732
Appendix F: Numerical optimization and solution of systems of equations 735
F.1 Maximization using Solver 735
F.2 The simplex method 740
F.3 Using Excel R° to solve equations 741
References 747

Library of Congress Subject Headings for this publication:

Insurance -- Statistical methods.
Insurance -- Mathematical models.