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Preface xiii PART I INTRODUCTION 1 Modeling 3 1.1 The model-based approach 3 1.2 Organization of this book 7 2 Random variables 11 2.1 Introduction 11 2.2 Key functions and four models 13 3 Basic distributional quantities 25 3.1 Moments 25 3.2 Percentiles 36 3.3 Generating functions and sums of random variables 38 3.4 Tails of distributions 41 3.5 Measures of Risk 50 PART II ACTUARIAL MODELS 4 Characteristics of Actuarial Models 63 4.1 Introduction 63 4.2 The role of parameters 65 5 Continuousmodels 77 5.1 Introduction 77 5.2 Creating new distributions 77 5.3 Selected distributions and their relationships 93 5.4 The linear exponential family 98 6 Discrete distributions 103 6.1 Introduction 103 6.2 The Poisson distribution 104 6.3 The negative binomial distribution 108 6.4 The binomial distribution 111 6.5 The (a, b, 0) class 113 6.6 Truncation and modification at zero 117 7 Advanced discrete distributions 125 7.1 Compound frequency distributions 125 7.2 Further properties of the compound Poisson class 133 7.3 Mixed frequency distributions 139 7.4 Effect of exposure on frequency 148 7.5 An inventory of discrete distributions 149 8 Frequency and severity with coverage modifications 153 8.1 Introduction 153 8.2 Deductibles 155 8.3 The loss elimination ratio and the effect of inflation for ordinary deductibles 161 8.4 Policy limits 164 8.5 Coinsurance, deductibles, and limits 167 8.6 The impact of deductibles on claim frequency 171 9 Aggregate loss models 179 9.1 Introduction 179 9.2 Model choices 184 9.3 The compound model for aggregate claims 186 9.4 Analytic results 203 9.5 Computing the aggregate claims distribution 209 9.6 The recursive method 211 9.7 The impact of individual policy modifications on aggregate payments 227 9.8 The individual risk model 232 PART III CONSTRUCTION OF EMPIRICAL MODELS 10 Review of mathematical statistics 245 10.1 Introduction 245 10.2 Point estimation 246 10.3 Interval estimation 257 10.4 Tests of hypotheses 260 11 Estimation for complete data 267 11.1 Introduction 267 11.2 The empirical distribution for complete, individual data 273 11.3 Empirical distributions for grouped data 278 12 Estimation for modified data 285 12.1 Point estimation 285 12.3 Kernel density models 308 12.4 Approximations for large data sets 314 PART IV PARAMETRIC STATISTICAL METHODS 13 Frequentist estimation 331 13.1 Method of moments and percentile matching 332 13.2 Maximum likelihood estimation 339 13.3 Variance and interval estimation 355 13.4 Non-normal confidence intervals 365 13.5 Maximum likelihood estimation of decrement probabilities 369 14 Frequentist Estimation for discrete distributions 373 14.1 Poisson 373 14.2 Negative binomial 378 14.3 Binomial 380 14.4 The (a, b, 1) class 384 14.5 Compound models 389 14.6 Effect of exposure on maximum likelihood estimation 391 14.7 Exercises 392 15 Bayesian estimation 397 15.1 Definitions and Bayes' theorem 398 15.2 Inference and prediction 402 15.3 Conjugate prior distributions and the linear exponential family 416 15.4 Computational issues 419 16 Model selection 421 16.1 Introduction 421 16.2 Representations of the data and model 422 16.3 Graphical comparison of the density and distribution functions 424 16.4 Hypothesis tests 430 16.5 Selecting a model 445 PART V CREDIBILITY 17 Introduction and Limited Fluctuation Credibility 467 17.1 Introduction 467 17.2 Limited fluctuation credibility theory 470 17.3 Full credibility 471 17.4 Partial credibility 475 17.5 Problems with the approach 480 17.6 Notes and References 480 17.7 Exercises 480 18 Greatest accuracy credibility 485 18.1 Introduction 485 18.2 Conditional distributions and expectation 489 18.3 The Bayesian methodology 494 18.4 The credibility premium 503 18.5 The Buhlmann model 507 18.6 The Buhlmann?Straub model 511 18.7 Exact credibility 518 18.8 Notes and References 522 18.9 Exercises 523 19 Empirical Bayes parameter estimation 541 19.1 Introduction 541 19.2 Nonparametric estimation 544 19.3 Semiparametric estimation
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