Nonlife actuarial models: theory, methodes and evaluation
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge
Cambridge Univ. Press
2009
|
| Edition: | 1. publ. |
| Series: | International series on actuarial science
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 518 - 520 |
| Physical Description: | XV, 524 S. graph. Darst. |
| ISBN: | 9780521764650 |
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| adam_text | Titel: Nonlife actuarial models
Autor: Tse, Yiu Kuen
Jahr: 2009
Contents
Preface PaKe x
Notation and convention xv
Part I Loss models 1
1 Claim-frequency distribution 3
1.1 Claim frequency, claim severity, and aggregate claim 4
1.2 Review of statistics 4
1.3 Some discrete distributions for claim frequency 6
1.3.1 Binomial distribution 7
1.3.2 Geometric distribution 8
1.3.3 Negative binomial distribution 9
1.3.4 Poisson distribution 11
1.4 The {a, b, 0) class of distributions 15
1.5 Some methods for creating new distributions 20
1.5.1 Compound distribution 21
1.5.2 Mixture distribution 31
1.6 Excel computation notes 34
1.7 Summary and conclusions 34
Exercises 36
2 Claim-severity distribution 41
2.1 Review of statistics 42
2.1.1 Survival function and hazard function 42
2.1.2 Mixed distribution 44
2.1.3 Expected value of function of random variable 45
2.1.4 Distribution of function of random variable 46
2.2 Some continuous distributions for claim severity 49
2.2.1 Exponential distribution 49
vn
viii Contents
2.2.2 Gamma distribution 50
2.2.3 Weibull distribution 51
2.2.4 Pareto distribution 51
2.3 Some methods for creating new distributions 52
2.3.1 Transformation of random variable 53
2.3.2 Mixture distribution 54
2.3.3 Splicing 58
2.4 Tail properties of claim severity 59
2.5 Effects of coverage modifications 66
2.5.1 Deductible 66
2.5.2 Policy limit 72
2.5.3 Coinsurance 73
2.5.4 Effects of inflation 76
2.5.5 Effects of deductible on claim frequency 77
2.6 Excel computation notes 79
2.7 Summary and conclusions 81
Exercises 82
3 Aggregate-loss models 86
3.1 Individual risk and collective risk models 87
3.2 Individual risk model 88
3.2.1 Exact distribution using convolution 89
3.2.2 Exact distribution using the De Pril recursion 92
3.2.3 Approximations of the individual risk model 94
3.3 Collective risk model 96
3.3.1 Properties of compound distributions 96
3.3.2 Panjer recursion 98
3.3.3 Approximations of the collective risk model 100
3.3.4 Compound Poisson distribution and individual
risk model 102
3.4 Coverage modifications and stop-loss reinsurance 103
3.5 Summary and conclusions 108
Exercises 108
Part II Risk and ruin 113
4 Risk measures 115
4.1 Uses of risk measures 116
4.2 Some premium-based risk measures 117
4.3 Axioms of coherent risk measures 118
4.4 Some capital-based risk measures 120
4.4.1 Value-at-Risk (VaR) 120
Contents ix
4.4.2 Conditional tail expectation and
related measures 123
4.5 More premium-based risk measures 129
4.5.1 Proportional hazard transform and risk-adjusted
premium 129
4.5.2 Esscher transform and risk-adjusted premium 132
4.6 Distortion-function approach 133
4.7 Wang transform 136
4.8 Summary and conclusions 138
Exercises 139
5 Ruin theory 143
5.1 Discrete-time surplus and events of ruin 144
5.2 Discrete-time ruin theory 145
5.2.1 Ultimate ruin in discrete time 146
5.2.2 Finite-time ruin in discrete time 150
5.2.3 Lundberg s inequality in discrete time 152
5.3 Continuous-time surplus function 157
5.4 Continuous-time ruin theory 159
5.4.1 Lundberg s inequality in continuous time 159
5.4.2 Distribution of deficit 163
5.5 Summary and conclusions 165
Exercises 165
Part III Credibility 169
6 Classical credibility 171
6.1 Framework and notations 172
6.2 Full credibility 173
6.2.1 Full credibility for claim frequency 173
6.2.2 Full credibility for claim severity 177
6.2.3 Full credibility for aggregate loss 179
6.2.4 Full credibility for pure premium 181
6.3 Partial credibility 182
6.4 Variation of assumptions 184
6.5 Summary and discussions 185
Exercises 186
7 Buhlmann credibility 190
7.1 Framework and notations 191
7.2 Variance components 192
7.3 Buhlmann credibility 201
7.4 Biihlmann-Straub credibility 208
x Contents
7.5 Summary and discussions 216
Exercises 217
8 Bayesian approach 223
8.1 Bayesian inference and estimation 224
8.1.1 Posterior distribution of parameter 225
8.1.2 Loss function and Bayesian estimation 228
8.1.3 Some examples of Bayesian credibility 230
8.2 Conjugate distributions 234
8.2.1 The gamma-Poisson conjugate distribution 235
8.2.2 The beta-geometric conjugate distribution 235
8.2.3 The gamma-exponential conjugate distribution 235
8.3 Bayesian versus Biihlmann credibility 235
8.4 Linear exponential family and exact credibility 242
8.5 Summary and discussions 248
Exercises 248
9 Empirical implementation of credibility 253
9.1 Empirical Bayes method 254
9.2 Nonparametric estimation 255
9.3 Semiparametric estimation 270
9.4 Parametric estimation 271
9.5 Summary and discussions 273
Exercises 274
Part IV Model construction and evaluation 279
10 Model estimation and types of data 281
10.1 Estimation 282
10.1.1 Parametric and nonparametric estimation 282
10.1.2 Point and interval estimation 282
10.1.3 Properties of estimators 283
10.2 Types of data 286
10.2.1 Duration data and loss data 286
10.2.2 Complete individual data 287
10.2.3 Incomplete individual data 289
10.2.4 Grouped data 294
10.3 Summary and discussions 296
Exercises 297
11 Nonparametric model estimation 301
11.1 Estimation with complete individual data 302
11.1.1 Empirical distribution 302
11.1.2 Kernel estimation of probability density function 306
Contents xi
11.2 Estimation with incomplete individual data 311
11.2.1 Kaplan-Meier (product-limit) estimator 311
11.2.2 Nelson-Aalen estimator 319
11.3 Estimation with grouped data 323
11.4 Excel computation notes 326
11.5 Summary and discussions 326
Exercises 327
12 Parametric model estimation 335
12.1 Methods of moments and percentile matching 336
12.1.1 Method of moments 336
12.1.2 Method of percentile matching 341
12.2 Bayesian estimation method 343
12.3 Maximum likelihood estimation method 344
12.3.1 Complete individual data 347
12.3.2 Grouped and incomplete data 351
12.4 Models with covariates 358
12.4.1 Proportional hazards model 358
12.4.2 Generalized linear model 364
12.4.3 Accelerated failure-time model 365
12.5 Modeling joint distribution using copula 366
12.6 Excel computation notes 369
12.7 Summary and discussions 371
Exercises 372
13 Model evaluation and selection 380
13.1 Graphical methods 381
13.2 Misspecification tests and diagnostic checks 385
13.2.1 Kolmogorov-Smirnov test 386
13.2.2 Anderson-Darling test 388
13.2.3 Chi-square goodness-of-fit test 389
13.2.4 Likelihood ratio test 391
13.3 Information criteria for model selection 393
13.4 Summary and discussions 394
Exercises 395
14 Basic Monte Carlo methods 400
14.1 Monte Carlo simulation 401
14.2 Uniform random number generators 402
14.3 General random number generators 405
14.3.1 Inversion method 406
14.3.2 Acceptance-rejection method 408
14.3.3 Generation of correlated random variables 411
xii Contents
14.4 Specific random number generators 414
14.4.1 Some continuous distributions 414
14.4.2 Some discrete distributions 417
14.5 Accuracy and Monte Carlo sample size 418
14.6 Variance reduction techniques 421
14.6.1 Antithetic variable 422
14.6.2 Control variable 423
14.6.3 Importance sampling 425
14.7 Excel computation notes 426
14.8 Summary and discussions 428
Exercises 428
15 Applications of Monte Carlo methods 435
15.1 Monte Carlo simulation for hypothesis test 436
15.1.1 Kolmogorov-Smirnov test 436
15.1.2 Chi-square goodness-of-fit test 438
15.2 Bootstrap estimation of p-value 439
15.3 Bootstrap estimation of bias and mean squared error 442
15.4 A general framework of bootstrap 445
15.5 Monte Carlo simulation of asset prices 447
15.5.1 Wiener process and generalized Wiener process 447
15.5.2 Diffusion process and lognormal distribution 448
15.5.3 Jump-diffusion process 453
15.6 Summary and discussions 455
Exercises 456
Appendix: Review of statistics 458
Answers to exercises 498
References 518
Index 521
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| indexdate | 2024-07-09T22:04:26Z |
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| isbn | 9780521764650 |
| language | English |
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| spelling | Tse, Yiu-Kuen Verfasser aut Nonlife actuarial models theory, methodes and evaluation Yiu-Kuen Tse 1. publ. Cambridge Cambridge Univ. Press 2009 XV, 524 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier International series on actuarial science Literaturverz. S. 518 - 520 Versicherungsmathematik (DE-588)4063194-1 gnd rswk-swf Versicherungsmathematik (DE-588)4063194-1 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018642070&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Tse, Yiu-Kuen Nonlife actuarial models theory, methodes and evaluation Versicherungsmathematik (DE-588)4063194-1 gnd |
| subject_GND | (DE-588)4063194-1 |
| title | Nonlife actuarial models theory, methodes and evaluation |
| title_auth | Nonlife actuarial models theory, methodes and evaluation |
| title_exact_search | Nonlife actuarial models theory, methodes and evaluation |
| title_full | Nonlife actuarial models theory, methodes and evaluation Yiu-Kuen Tse |
| title_fullStr | Nonlife actuarial models theory, methodes and evaluation Yiu-Kuen Tse |
| title_full_unstemmed | Nonlife actuarial models theory, methodes and evaluation Yiu-Kuen Tse |
| title_short | Nonlife actuarial models |
| title_sort | nonlife actuarial models theory methodes and evaluation |
| title_sub | theory, methodes and evaluation |
| topic | Versicherungsmathematik (DE-588)4063194-1 gnd |
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