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| Preface | |
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| Introduction | |
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| Modeling | |
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| The model-based approach | |
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| Organization of this book | |
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| Random variables | |
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| Introduction | |
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| Key functions and four models | |
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| Basic distributional quantities | |
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| Moments | |
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| Percentiles | |
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| Generating functions and sums of random variables | |
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| Tails of distributions | |
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| Measures of Risk | |
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| Actuarial Models | |
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| Characteristics of Actuarial Models | |
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| Introduction | |
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| The role of parameters | |
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| Continuousmodels | |
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| Introduction | |
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| Creating new distributions | |
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| Selected distributions and their relationships | |
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| The linear exponential family | |
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| Discrete distributions | |
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| Introduction | |
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| The Poisson distribution | |
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| The negative binomial distribution | |
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| The binomial distribution | |
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| The (a, b, 0) class | |
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| Truncation and modification at zero | |
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| Advanced discrete distributions | |
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| Compound frequency distributions | |
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| Further properties of the compound Poisson class | |
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| Mixed frequency distributions | |
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| Effect of exposure on frequency | |
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| An inventory of discrete distributions | |
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| Frequency and severity with coverage modifications | |
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| Introduction | |
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| Deductibles | |
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| The loss elimination ratio and the effect of inflation for ordinary deductibles | |
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| Policy limits | |
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| Coinsurance, deductibles, and limits | |
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| The impact of deductibles on claim frequency | |
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| Aggregate loss models | |
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| Introduction | |
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| Model choices | |
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| The compound model for aggregate claims | |
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| Analytic results | |
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| Computing the aggregate claims distribution | |
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| The recursive method | |
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| The impact of individual policy modifications on aggregate payments | |
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| The individual risk model | |
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| Construction Of Empirical Models | |
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| Review of mathematical statistics | |
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| Introduction | |
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| Point estimation | |
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| Interval estimation | |
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| Tests of hypotheses | |
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| Estimation for complete data | |
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| Introduction | |
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| The empirical distribution for complete, individual data | |
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| Empirical distributions for grouped data | |
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| Estimation for modified data | |
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| Point estimation | |
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| Kernel density models | |
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| Approximations for large data sets | |
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| Parametric Statistical Methods | |
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| Frequentist estimation | |
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| Method of moments and percentile matching | |
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| Maximum likelihood estimation | |
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| Variance and interval estimation | |
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| Non-normal confidence intervals | |
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| Maximum likelihood estimation of decrement probabilities | |
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| Frequentist Estimation for discrete distributions | |
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| Poisson | |
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| Negative binomial | |
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| Binomial | |
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| The (a, b, 1) class | |
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| Compound models | |
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| Effect of exposure on maximum likelihood estimation | |
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| Exercises | |
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| Bayesian estimation | |
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| Definitions and Bayes' theorem | |
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| Inference and prediction | |
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| Conjugate prior distributions and the linear exponential family | |
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| Computational issues | |
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| Model selection | |
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| Introduction | |
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| Representations of the data and model | |
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| Graphical comparison of the density and distribution functions | |
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| Hypothesis tests | |
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| Selecting a model | |
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| Credibility | |
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| Introduction and Limited Fluctuation Credibility | |
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| Introduction | |
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| Limited fluctuation credibility theory | |
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| Full credibility | |
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| Partial credibility | |
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| Problems with the approach | |
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| Notes and References | |
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| Exercises | |
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| Greatest accuracy credibility | |
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| Introduction | |
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| Conditional distributions and expectation | |
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| The Bayesian methodology | |
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| The credibility premium | |
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| The Buhlmann model | |
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| The Buhlmann?Straub model | |
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| Exact credibility | |
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| Notes and References | |
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| Exercises | |
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| Empirical Bayes parameter estimation | |
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| Introduction | |
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| Nonparametric estimation | |
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| Semiparametric estimation | |
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| Notes and References | |
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| Exercises | |
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| Simulation | |
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| Simulation | |
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| Basics of simulation | |
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| Simulation for specific distributions | |
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| Determining the sample size | |
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| Examples of simulation in actuarial modeling | |
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| An inventory of continuous distributions | |
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| Introduction | |
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| Transformed beta family | |
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| Transformed gamma family | |
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| Distributions for large losses | |
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| Other distributions | |
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| Distributions with finite support | |
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| An inventory of discrete distributions | |
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| Introduction | |
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| The (a , b , 0) class | |
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| The (a , b , 1) class | |
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| The compound class | |
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| A hierarchy of discrete distributions | |
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| Frequency and severity relationships | |
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| The recursive formula | |
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| Discretization of the severity distribution | |
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| The method of rounding | |
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| Mean preserving | |
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| Undiscretization of a discretized distribution | |
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| Numerical optimization and solution of systems of equations | |
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| Maximization using Solver | |
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| The simplex method | |
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| Using Excel� to solve equations | |
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| References | |