Statistical and Probabilistic Methods in

Statistical and Probabilistic Methods in Actuarial Sciencecovers 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.Â 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	p. v
Preface	p. vii
Introduction	p. ix
Claims Reserving and Pricing with Run-Off Triangles	p. 1
The evolving nature of claims and reserves	p. 1
Chain ladder methods	p. 4
Basic chain ladder method	p. 5
Inflation-adjusted chain ladder method	p. 8
The average cost per claim method	p. 11
The Bornhuetter-Ferguson or loss ratio method	p. 14
An example in pricing products	p. 19
Statistical modeling and the separation technique	p. 26
Problems	p. 27
Loss Distributions	p. 35
Introduction to loss distributions	p. 35
Classical loss distributions	p. 36
Exponential distribution	p. 36
Pareto distribution	p. 39
Gamma distribution	p. 43
Weibull distribution	p. 45
Lognormal distribution	p. 47
Fitting loss distributions	p. 51
Kolmogorov-Smirnoff test	p. 52
Chi-square goodness-of-fit tests	p. 54
Akaike information criteria	p. 58
Mixture distributions	p. 58
Loss distributions and reinsurance	p. 61
Proportional reinsurance	p. 62
Excess of loss reinsurance	p. 62
Problems	p. 68
Risk Theory	p. 77
Risk models for aggregate claims	p. 77
Collective risk models	p. 78
Basic properties of compound distributions	p. 79
Compound Poisson, binomial and negative binomial distributions	p. 79
Sums of compound Poisson distributions	p. 85
Exact expressions for the distribution of S	p. 87
Approximations for the distribution of S	p. 92
Individual risk models for S	p. 94
Basic properties of the individual risk model	p. 95
Compound binomial distributions and individual risk models	p. 97
Compound Poisson approximations for individual risk models	p. 98
Premiums and reserves for aggregate claims	p. 99
Determining premiums for aggregate claims	p. 99
Setting aside reserves for aggregate claims	p. 103
Reinsurance for aggregate claims	p. 107
Proportional reinsurance	p. 109
Excess of loss reinsurance	p. 111
Stop-loss reinsurance	p. 116
Problems	p. 120
Ruin Theory	p. 129
The probability of ruin in a surplus process	p. 129
Surplus and aggregate claims processes	p. 129
Probability of ruin in discrete time	p. 132
Poisson surplus processes	p. 132
Probability of ruin and the adjustment coefficient	p. 134
The adjustment equation	p. 135
Lundberg's bound on the probability of ruin [psi] (U)	p. 138
The probability of ruin when claims are exponentially distributed	p. 140
Reinsurance and the probability of ruin	p. 146
Adjustment coefficients and proportional reinsurance	p. 147
Adjustment coefficients and excess of loss reinsurance	p. 149
Problems	p. 152
Credibility Theory	p. 159
Introduction to credibility estimates	p. 159
Classical credibility theory	p. 161
Full credibility	p. 161
Partial credibility	p. 163
The Bayesian approach to credibility theory	p. 164
Bayesian credibility	p. 164
Greatest accuracy credibility theory	p. 170
Bayes and linear estimates of the posterior mean	p. 172
Predictive distribution for X[subscript n+] 1	p. 175
Empirical Bayes approach to credibility theory	p. 176
Empirical Bayes credibility - Model 1	p. 177
Empirical Bayes credibility - Model 2	p. 180
Problems	p. 183
No Claim Discounting in Motor Insurance	p. 191
Introduction to No Claim Discount schemes	p. 191
Transition in a No Claim Discount system	p. 193
Discount classes and movement in NCD schemes	p. 193
One-step transition probabilities in NCD schemes	p. 195
Limiting distributions and stability in NCD models	p. 198
Propensity to make a claim in NCD schemes	p. 204
Thresholds for claims when an accident occurs	p. 205
The claims rate process in an NCD system	p. 208
Reducing heterogeneity with NCD schemes	p. 212
Problems	p. 214
Generalized Linear Models	p. 221
Introduction to linear and generalized linear models	p. 221
Multiple linear regression and the normal model	p. 225
The structure of generalized linear models	p. 230
Exponential families	p. 232
Link functions and linear predictors	p. 236
Factors and covariates	p. 238
Interactions	p. 238
Minimally sufficient statistics	p. 244
Model selection and deviance	p. 245
Deviance and the saturated model	p. 245
Comparing models with deviance	p. 248
Residual analysis for generalized linear models	p. 252
Problems	p. 258
Decision and Game Theory	p. 265
Introduction	p. 265
Game theory	p. 267
Zero-sum two-person games	p. 268
Minimax and saddle point strategies	p. 270
Randomized strategies	p. 273
The Prisoner's Dilemma and Nash equilibrium in variable-sum games	p. 278
Decision making and risk	p. 280
The minimax criterion	p. 283
The Bayes criterion	p. 283
Utility and expected monetary gain	p. 288
Rewards, prospects and utility	p. 290
Utility and insurance	p. 292
Problems	p. 295
References	p. 304
Basic Probability Distributions	p. 309
Some Basic Tools in Probability and Statistics	p. 313
Moment generating functions	p. 313
Convolutions of random variables	p. 316
Conditional probability and distributions	p. 317
The double expectation theorem and E(X)	p. 319
The random variable V(X/Y)	p. 322
Maximum likelihood estimation	p. 324
An Introduction to Bayesian Statistics	p. 327
Bayesian statistics	p. 327
Conjugate families	p. 328
Loss functions and Bayesian inference	p. 329
Answers to Selected Problems	p. 335
Claims reserving and pricing with run-off triangles	p. 335
Loss distributions	p. 335
Risk theory	p. 337
Ruin theory	p. 338
Credibility theory	p. 338
No claim discounting in motor insurance	p. 340
Generalized linear models	p. 340
Decision and game theory	p. 341
Index	p. 345
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Statistical and Probabilistic Methods in Actuarial Science

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