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E-grāmata: Statistical and Probabilistic Methods in Actuarial Science [Taylor & Francis e-book]

(University College Dublin, Ireland)
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Statistical and Probabilistic Methods in Actuarial SciencePalielināt
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Statistical and Probabilistic Methods in Actuarial Science covers many of the diverse methods in applied probability and statistics for students aspiring to careers in insurance, actuarial science, and finance. The book builds on students existing knowledge of probability and statistics by establishing a solid and thorough understanding of these methods. It also emphasizes the wide variety of practical situations in insurance and actuarial science where these techniques may be used.

Although some chapters are linked, several can be studied independently from the others. The first chapter introduces claims reserving via the deterministic chain ladder technique. The next few chapters survey loss distributions, risk models in a fixed period of time, and surplus processes, followed by an examination of credibility theory in which collateral and sample information are brought together to provide reasonable methods of estimation. In the subsequent chapter, experience rating via no claim discount schemes for motor insurance provides an interesting application of Markov chain methods. The final chapters discuss generalized linear models and decision and game theory.

Developed by an author with many years of teaching experience, this text presents an accessible, sound foundation in both the theory and applications of actuarial science. It encourages students to use the statistical software package R to check examples and solve problems.
Dedication v
Preface vii
Introduction ix
1 Claims Reserving and Pricing with Run-Off Triangles 1
1.1 The evolving nature of claims and reserves
1
1.2 Chain ladder methods
4
1.2.1 Basic chain ladder method
5
1.2.2 Inflation-adjusted chain ladder method
8
1.3 The average cost per claim method
11
1.4 The Bornhuetter—Ferguson or loss ratio method
14
1.5 An example in pricing products
19
1.6 Statistical modeling and the separation technique
26
1.7 Problems
27
2 Loss Distributions 35
2.1 Introduction to loss distributions
35
2.2 Classical loss distributions
36
2.2.1 Exponential distribution
36
2.2.2 Pareto distribution
39
2.2.3 Gamma distribution
43
2.2.4 Weibull distribution
45
2.2.5 Lognormal distribution
47
2.3 Fitting loss distributions
51
2.3.1 Kolmogorov -Smirnoff test
52
2.3.2 Chi-square goodness-of-fit tests
54
2.3.3 Akaike information criteria
58
2.4 Mixture distributions
58
2.5 Loss distributions and reinsurance
61
2.5.1 Proportional reinsurance
62
2.5.2 Excess of loss reinsurance
62
2.6 Problems
68
3 Risk Theory 77
3.1 Risk models for aggregate claims
77
3.2 Collective risk models
78
3.2.1 Basic properties of compound distributions
79
3.2.2 Compound Poisson, binomial and negative binomial distributions
79
3.2.3 Sums of compound Poisson distributions
85
3.2.4 Exact expressions for the distribution of S
87
3.2.5 Approximations for the distribution of S
92
3.3 Individual risk models for S
94
3.3.1 Basic properties of the individual risk model
95
3.3.2 Compound binomial distributions and individual risk models
97
3.3.3 Compound Poisson approximations for individual risk models
98
3.4 Premiums and reserves for aggregate claims
99
3.4.1 Determining premiums for aggregate claims
99
3.4.2 Setting aside reserves for aggregate claims
103
3.5 Reinsurance for aggregate claims
107
3.5.1 Proportional reinsurance
109
3.5.2 Excess of loss reinsurance
111
3.5.3 Stop-loss reinsurance
116
3.6 Problems
120
4 Ruin Theory 129
4.1 The probability of ruin in a surplus process
129
4.2 Surplus and aggregate claims processes
129
4.2.1 Probability of ruin in discrete time
132
4.2.2 Poisson surplus processes
132
4.3 Probability of ruin and the adjustment coefficient
134
4.3.1 The adjustment equation
135
4.3.2 Lundberg's bound on the probability of ruin ψ(U)
138
4.3.3 The probability of ruin when claims are exponentially distributed
140
4.4 Reinsurance and the probability of ruin
146
4.4.1 Adjustment coefficients and proportional reinsurance
147
4.4.2 Adjustment coefficients and excess of loss reinsurance
149
4.5 Problems
152
5 Credibility Theory 159
5.1 Introduction to credibility estimates
159
5.2 Classical credibility theory
161
5.2.1 Full credibility
161
5.2.2 Partial credibility
163
5.3 The Bayesian approach to credibility theory
164
5.3.1 Bayesian credibility
164
5.4 Greatest accuracy credibility theory
170
5.4.1 Bayes and linear estimates of the posterior mean
172
5.4.2 Predictive distribution for Xn+1
175
5.5 Empirical Bayes approach to credibility theory
176
5.5.1 Empirical Bayes credibility – Model 1
177
5.5.2 Empirical Bayes credibility – Model 2
180
5.6 Problems
183
6 No Claim Discounting in Motor Insurance 191
6.1 Introduction to No Claim Discount schemes
191
6.2 Transition in a No Claim Discount system
193
6.2.1 Discount classes and movement in NCD schemes
193
6.2.2 One-step transition probabilities in NCD schemes
195
6.2.3 Limiting distributions and stability in NCD models
198
6.3 Propensity to make a claim in NCD schemes
204
6.3.1 Thresholds for claims when an accident occurs
205
6.3.2 The claims rate process in an NCD system
208
6.4 Reducing heterogeneity with NCD schemes
212
6.5 Problems
214
7 Generalized Linear Models 221
7.1 Introduction to linear and generalized linear models
221
7.2 Multiple linear regression and the normal model
225
7.3 The structure of generalized linear models
230
7.3.1 Exponential families
232
7.3.2 Link functions and linear predictors
236
7.3.3 Factors and covariates
238
7.3.4 Interactions
238
7.3.5 Minimally sufficient statistics
244
7.4 Model selection and deviance
245
7.4.1 Deviance and the saturated model
245
7.4.2 Comparing models with deviance
248
7.4.3 Residual analysis for generalized linear models
252
7.5 Problems
258
8 Decision and Game Theory 265
8.1 Introduction
265
8.2 Game theory
267
8.2.1 Zero-sum two-person games
268
8.2.2 Minimax and saddle point strategies
270
8.2.3 Randomized strategies
273
8.2.4 The Prisoner's Dilemma and Nash equilibrium in variable-sum games
278
8.3 Decision making and risk
280
8.3.1 The minimax criterion
283
8.3.2 The Bayes criterion
283
8.4 Utility and expected monetary gain
288
8.4.1 Rewards, prospects and utility
290
8.4.2 Utility and insurance
292
8.5 Problems
295
References 304
Appendix A Basic Probability Distributions 309
Appendix B Some Basic Tools in Probability and Statistics 313
B.1 Moment generating functions
313
B.2 Convolutions of random variables
316
B.3 Conditional probability and distributions
317
B.3.1 The double expectation theorem and E(X)
319
B.3.2 The random variable V(X|Y)
322
B.4 Maximum likelihood estimation
324
Appendix C An Introduction to Bayesian Statistics 327
C.1 Bayesian statistics
327
C.1.1 Conjugate families
328
C.1.2 Loss functions and Bayesian inference
329
Appendix D Answers to Selected Problems 335
D.1 Claims reserving and pricing with run-off triangles
335
D.2 Loss distributions
335
D.3 Risk theory
337
D.4 Ruin theory
338
D.5 Credibility theory
338
D.6 No claim discounting in motor insurance
340
D.7 Generalized linear models
340
D.8 Decision and game theory
341
Index 345


University College Dublin, Ireland
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