Statistical Methods in Insurance and Actuarial Science Book

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