Loss models: from data to decisions
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Hoboken, NJ
Wiley
2012
|
| Ausgabe: | 4. ed. |
| Schriftenreihe: | Wiley series in probability and statistics
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XIV, 512 S. graph. Darst. |
| ISBN: | 9781118315323 |
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Datensatz im Suchindex
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|---|---|
| adam_text | Titel: Loss models
Autor: Klugman, Stuart A
Jahr: 2012
CONTENTS
Preface xiii
PART I INTRODUCTION
1 Modeling 3
1.1 The model-based approach 3
1.1.1 The modeling process 4
1.1.2 The modeling advantage 5
1.2 Organization of this book 5
2 Random variables 7
2.1 Introduction 7
2.2 Key functions and four models 9
2.2.1 Exercises 17
3 Basic distributional quantities 19
3.1 Moments 19
3.1.1 Exercises 26
3.2 Percentiles 27
3.2.1 Exercises 28
3.3 Generating functions and sums of random variables 29
V
vi CONTENTS
3.3.1 Exercises 30
3.4 Tails of distributions 31
3.4.1 Classification based on moments 31
3.4.2 Comparison based on limiting tail behavior 32
3.4.3 Classification based on the hazard rate function 33
3.4.4 Classification based on the mean excess loss function 34
3.4.5 Equilibrium distributions and tail behavior 36
3.4.6 Exercises 37
3.5 Measures of Risk 38
3.5.1 Introduction 38
3.5.2 Risk measures and coherence 39
3.5.3 Value-at-Risk 40
3.5.4 Tail-Value-at-Risk 42
3.5.5 Exercises 46
PART II ACTUARIAL MODELS
4 Characteristics of Actuarial Models 49
4.1 Introduction 49
4.2 The role of parameters 49
4.2.1 Parametric and scale distributions 50
4.2.2 Parametric distribution families 52
4.2.3 Finite mixture distributions 52
4.2.4 Data-dependent distributions 55
4.2.5 Exercises 56
5 Continuous models 59
5.1 Introduction 59
5.2 Creating new distributions 59
5.2.1 Multiplication by a constant 60
5.2.2 Raising to a power 60
5.2.3 Exponentiation 62
5.2.4 Mixing 62
5.2.5 Frailty models 65
5.2.6 Splicing 66
5.2.7 Exercises 68
5.3 Selected distributions and their relationships 72
5.3.1 Introduction 72
5.3.2 Two parametric families 72
5.3.3 Limiting distributions 72
5.3.4 Two heavy-tailed distributions 74
5.3.5 Exercises 75
CONTENTS Vii
5.4 The linear exponential family 75
5.4.1 Exercises 78
6 Discrete distributions 79
6.1 Introduction 79
6.1.1 Exercise 80
6.2 The Poisson distribution 80
6.3 The negative binomial distribution 83
6.4 The binomial distribution 85
6.5 The (a, b, 0) class 86
6.5.1 Exercises 89
6.6 Truncation and modification at zero 89
6.6.1 Exercises 94
7 Advanced discrete distributions 95
7.1 Compound frequency distributions 95
7.1.1 Exercises 101
7.2 Further properties of the compound Poisson class 101
7.2.1 Exercises 107
7.3 Mixed frequency distributions 107
7.3.1 General mixed frequency distribution 107
7.3.2 Mixed Poisson distributions 109
7.3.3 Exercises 113
7.4 Effect of exposure on frequency 114
Appendix: An inventory of discrete distributions 114
A.0.1 Exercises 115
8 Frequency and severity with coverage modifications 117
8.1 Introduction 117
8.2 Deductibles 117
8.2.1 Exercises 122
8.3 The loss elimination ratio and the effect of inflation for ordinary
deductibles 122
8.3.1 Exercises 124
8.4 Policy limits 125
8.4.1 Exercises 127
8.5 Co insurance, deductibles, and limits 127
8.5.1 Exercises 129
8.6 The impact of deductibles on claim frequency 131
8.6.1 Exercises 134
viii CONTENTS
9 Aggregate loss models 137
9.1 Introduction 137
9.1.1 Exercises 140
9.2 Model choices 140
9.2.1 Exercises 141
9.3 The compound model for aggregate claims 141
9.3.1 Exercises 148
9.4 Analytic results 155
9.4.1 Exercises 158
9.5 Computing the aggregate claims distribution 159
9.6 The recursive method 161
9.6.1 Applications to compound frequency models 163
9.6.2 Underflow/overflow problems 165
9.6.3 Numerical stability 165
9.6.4 Continuous severity 166
9.6.5 Constructing arithmetic distributions 166
9.6.6 Exercises 169
9.7 The impact of individual policy modifications on aggregate payments 173
9.7.1 Exercises 176
9.8 The individual risk model 176
9.8.1 The model 176
9.8.2 Parametric approximation 178
9.8.3 Compound Poisson approximation 180
9.8.4 Exercises 182
PART III CONSTRUCTION OF EMPIRICAL MODELS
10 Review of mathematical statistics 187
10.1 Introduction 187
10.2 Point estimation 188
10.2.1 Introduction 188
10.2.2 Measures of quality 189
10.2.3 Exercises 195
10.3 Interval estimation 196
10.3.1 Exercises 198
10.4 Tests of hypotheses 198
10.4.1 Exercise 202
11 Estimation for complete data 203
11.1 Introduction 203
11.2 The empirical distribution for complete, individual data 207
CONTENTS iX
11.2.1 Exercises 211
11.3 Empirical distributions for grouped data 211
11.3.1 Exercises 214
12 Estimation for modified data 217
12.1 Point estimation 217
12.1.1 Exercises 224
12.2 Means, variances, and interval estimation 225
12.2.1 Exercises 234
12.3 Kernel density models 236
12.3.1 Exercises 239
12.4 Approximations for large data sets 240
12.4.1 Introduction 240
12.4.2 Using individual data points 242
12.4.3 Interval-based methods 245
12.4.4 Exercises 249
PART IV PARAMETRIC STATISTICAL METHODS
13 Frequentisi estimation 253
13.1 Method of moments and percentile matching 253
13.1.1 Exercises 257
13.2 Maximum likelihood estimation 259
13.2.1 Introduction 259
13.2.2 Complete, individual data 261
13.2.3 Complete, grouped data 262
13.2.4 Truncated or censored data 263
13.2.5 Exercises 266
13.3 Variance and interval estimation 272
13.3.1 Exercises 278
13.4 Nonnormal confidence intervals 280
13.4.1 Exercise 282
13.5 Maximum likelihood estimation of decrement probabilities 282
13.5.1 Exercise 284
14 Frequentisi Estimation for discrete distributions 285
14.1 Poisson 285
14.2 Negative binomial 289
14.3 Binomial 291
14.4 The (a, b, 1) class 293
14.5 Compound models 297
14.6 Effect of exposure on maximum likelihood estimation 299
X
CONTENTS
14.7 Exercises 300
15 Bayesian estimation 305
15.1 Definitions and Bayes Theorem 305
15.2 Inference and prediction 309
15.2.1 Exercises 315
15.3 Conjugate prior distributions and the linear exponential family 320
15.3.1 Exercises 321
15.4 Computational issues 322
16 Model selection 323
16.1 Introduction 323
16.2 Representations of the data and model 324
16.3 Graphical comparison of the density and distribution functions 325
16.3.1 Exercises 330
16.4 Hypothesis tests 330
16.4.1 Kolmogorov-Smirnov test 330
16.4.2 Anderson-Darling test 332
16.4.3 Chi-square goodness-of-fit test 333
16.4.4 Likelihood ratio test 337
16.4.5 Exercises 339
16.5 Selecting a model 342
16.5.1 Introduction 342
16.5.2 Judgment-based approaches 342
16.5.3 Score-based approaches 343
16.5.4 Exercises 350
PART V CREDIBILITY
17 Introduction and Limited Fluctuation Credibility 357
17.1 Introduction 357
17.2 Limited fluctuation credibility theory 359
17.3 Full credibility 360
17.4 Partial credibility 363
17.5 Problems with the approach 366
17.6 Notes and References 367
17.7 Exercises 367
18 Greatest accuracy credibility 371
18.1 Introduction 371
18.2 Conditional distributions and expectation 373
18.3 The Bayesian methodology 377
CONTENTS Xi
18.4 The credibility premium 385
18.5 The Bühlmann model 388
18.6 The Biihlmann-Straub model 392
18.7 Exact credibility 397
18.8 Notes and References 401
18.9 Exercises 402
19 Empirical Bayes parameter estimation 415
19.1 Introduction 415
19.2 Nonparametric estimation 418
19.3 Semiparametric estimation 428
19.4 Notes and References 430
19.5 Exercises 430
PART VI SIMULATION
20 Simulation 437
20.1 Basics of simulation 437
20.1.1 The simulation approach 438
20.1.2 Exercises 442
20.2 Simulation for specific distributions 442
20.2.1 Discrete mixtures 442
20.2.2 Time or age of death from a life table 443
20.2.3 Simulating from the (a, b, 0) class 444
20.2.4 Normal and lognormal distributions 446
20.2.5 Exercises 447
20.3 Determining the sample size 448
20.3.1 Exercises 449
20.4 Examples of simulation in actuarial modeling 450
20.4.1 Aggregate loss calculations 450
20.4.2 Examples of lack of independence 450
20.4.3 Simulation analysis of the two examples 451
20.4.4 Using simulation to determine risk measures 454
20.4.5 Statistical analyses 454
20.4.6 Exercises 456
A An inventory of continuous distributions 459
A. 1 Introduction 459
A.2 Transformed beta family 463
A.2.1 Four-parameter distribution 463
A.2.2 Three-parameter distributions 463
A.2.3 Two-parameter distributions 465
xii CONTENTS
A.3 Transformed gamma family 467
A.3.1 Three-parameter distributions 467
A.3.2 Two-parameter distributions 468
A.3.3 One-parameter distributions 469
A.4 Distributions for large losses 470
A.4.1 Extreme value distributions 470
A.4.2 Generalized Pareto distributions 471
A.5 Other distributions 471
A.6 Distributions with finite support 473
B An inventory of discrete distributions 475
B.l Introduction 475
B.2 The (a, b, 0) class 476
B.3 The (a, b, 1) class 477
B.3.1 The zero-truncated subclass 477
B.3.2 The zero-modified subclass 479
B.4 The compound class 480
B.4.1 Some compound distributions 480
B.5 A hierarchy of discrete distributions 482
C Frequency and severity relationships 483
D The recursive formula 485
E Discretization of the severity distribution 487
E. 1 The method of rounding 487
E.2 Mean preserving 488
E.3 Undiscretization of a discretized distribution 488
F Numerical optimization and solution of systems of equations 491
F.l Maximization using Solver 491
F.2 The simplex method 495
F.3 Using Excel® to solve equations 496
References 501
Index
507
|
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| author | Klugman, Stuart A. 1949- Panjer, Harry H. 1946- Willmot, Gordon E. 1957- |
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| series2 | Wiley series in probability and statistics |
| spelling | Klugman, Stuart A. 1949- Verfasser (DE-588)1044735155 aut Loss models from data to decisions Stuart A. Klugman ; Harry H. Panjer ; Gordon E. Willmot 4. ed. Hoboken, NJ Wiley 2012 XIV, 512 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Wiley series in probability and statistics Verlustrechnung (DE-588)4187883-8 gnd rswk-swf Mathematisches Modell (DE-588)4114528-8 gnd rswk-swf Versicherungsmathematik (DE-588)4063194-1 gnd rswk-swf Versicherungsstatistik (DE-588)4235273-3 gnd rswk-swf Risikotheorie (DE-588)4135592-1 gnd rswk-swf (DE-588)4143389-0 Aufgabensammlung gnd-content Versicherungsmathematik (DE-588)4063194-1 s Verlustrechnung (DE-588)4187883-8 s Mathematisches Modell (DE-588)4114528-8 s DE-604 Risikotheorie (DE-588)4135592-1 s Versicherungsstatistik (DE-588)4235273-3 s Panjer, Harry H. 1946- Verfasser (DE-588)1044735732 aut Willmot, Gordon E. 1957- Verfasser (DE-588)122585860 aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025729750&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Klugman, Stuart A. 1949- Panjer, Harry H. 1946- Willmot, Gordon E. 1957- Loss models from data to decisions Verlustrechnung (DE-588)4187883-8 gnd Mathematisches Modell (DE-588)4114528-8 gnd Versicherungsmathematik (DE-588)4063194-1 gnd Versicherungsstatistik (DE-588)4235273-3 gnd Risikotheorie (DE-588)4135592-1 gnd |
| subject_GND | (DE-588)4187883-8 (DE-588)4114528-8 (DE-588)4063194-1 (DE-588)4235273-3 (DE-588)4135592-1 (DE-588)4143389-0 |
| title | Loss models from data to decisions |
| title_auth | Loss models from data to decisions |
| title_exact_search | Loss models from data to decisions |
| title_full | Loss models from data to decisions Stuart A. Klugman ; Harry H. Panjer ; Gordon E. Willmot |
| title_fullStr | Loss models from data to decisions Stuart A. Klugman ; Harry H. Panjer ; Gordon E. Willmot |
| title_full_unstemmed | Loss models from data to decisions Stuart A. Klugman ; Harry H. Panjer ; Gordon E. Willmot |
| title_short | Loss models |
| title_sort | loss models from data to decisions |
| title_sub | from data to decisions |
| topic | Verlustrechnung (DE-588)4187883-8 gnd Mathematisches Modell (DE-588)4114528-8 gnd Versicherungsmathematik (DE-588)4063194-1 gnd Versicherungsstatistik (DE-588)4235273-3 gnd Risikotheorie (DE-588)4135592-1 gnd |
| topic_facet | Verlustrechnung Mathematisches Modell Versicherungsmathematik Versicherungsstatistik Risikotheorie Aufgabensammlung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025729750&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT klugmanstuarta lossmodelsfromdatatodecisions AT panjerharryh lossmodelsfromdatatodecisions AT willmotgordone lossmodelsfromdatatodecisions |