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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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Quantiles | |
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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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Continuous models | |
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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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TVaR for continuous distributions | |
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Extreme value distributions | |
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Discrete distributions and processes | |
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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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Counting processes | |
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Truncation and modification at zero | |
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Compound frequency models | |
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Further properties of the compound Poisson class | |
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Mixed Poisson distributions | |
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Mixed Poisson processes | |
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Effect of exposure on frequency | |
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An inventory of discrete distributions | |
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TVaR for discrete distributions | |
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Multivariate models | |
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Introduction | |
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Sklar�s theorem and copulas | |
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Measures of dependency | |
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Tail dependence | |
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Archimedean copulas | |
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Elliptical copulas | |
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Extreme value copulas | |
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Archimax copulas | |
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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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Inversion methods | |
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Calculations with approximate distributions | |
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Comparison of methods | |
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The individual risk model | |
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TVaR for aggregate losses | |
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Discrete-time ruin models | |
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Introduction | |
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Process models for insurance | |
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Discrete, finite-time ruin probabilities | |
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Continuous-time ruin models | |
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Introduction | |
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The adjustment coefficient and Lundberg�s inequality | |
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An integrodifferential equation | |
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The maximum aggregate loss | |
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Cramer�s asymptotic ruin formula and Tijms' approximation | |
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The Brownian motion risk process | |
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Brownian motion and the probability of ruin | |
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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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Means, variances, and interval 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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Parameter 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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Bayesian estimation | |
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Estimation for discrete distributions | |
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Exercises | |
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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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Estimation and model selection for more complex models | |
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Extreme value models | |
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Copula models | |
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Models with covariates | |
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Five examples | |
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Introduction | |
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Time to death | |
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Time from incidence to report | |
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Payment amount | |
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An aggregate loss example | |
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Another aggregate loss example | |
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Comprehensive exercises | |
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Adjusted Estimates | |
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Interpolation and smoothing | |
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Introduction | |
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Polynomial interpolation and smoothing | |
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Cubic spline interpolation | |
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Approximating functions with splines | |
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Extrapolating with splines | |
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Smoothing splines | |
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Credibility | |
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Introduction | |
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Limited fluctuation credibility theory | |
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Greatest accuracy credibility theory | |
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Empirical Bayes parameter estimation | |
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Simulation | |
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Simulation | |
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Basics of simulation | |
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Examples of simulation in actuarial modeling | |
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Examples of simulation in finance | |
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An inventory of continuous distributions | |
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An inventory 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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Numerical optimization and solution of systems of equations | |
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References | |
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Index | |