583 lines
10 KiB
Markdown
583 lines
10 KiB
Markdown
---
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title: "Loss Models: From Data to Decisions, Fifth Edition"
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tags:
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- authorship/other
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- exclude-from-word-count
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- topic/risk
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- type/media/book
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authors:
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- Gordon E. Willmot
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- Harry H. Panjer
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- Stuart A. Klugman
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edition: Fifth
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publisher: John Wiley & Sons, Inc.
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series: Wiley Series in Probability and Statistics
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subtitle: From Data to Decisions
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type: book
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year: 2019
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---
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# Loss Models: From Data to Decisions, Fifth Edition
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%%
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This note, with the exception of comments like this one
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(reserved for notes on transcription)
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consists only of content from the text.
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For commentary see the companion [[loss-models]].
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%%
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## Part I Introduction
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### 1. Modeling
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#### 1.1 The Model-Based Approach
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##### 1.1.1 The Modeling Process
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The model-based approach
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should be considered in the context of the objectives of any given problem.
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Many problems in actuarial science
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involve the building of a mathematical model
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that can be used to forecast or predict insurance costs in the future.
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##### 1.1.2 The Modeling Advantage
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#### 1.2 The Organization of This Book
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### 2. Random Variables
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#### 2.1 Introduction
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#### 2.2 Key Functions and Four Models
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##### 2.2.1 Exercises
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### 3. Basic Distributional Quantities
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#### 3.1 Moments
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##### 3.1.1 Exercises
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#### 3.2 Percentiles
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##### 3.2.1 Exercises
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#### 3.3 Generating Functions and Sums of Random Variables
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##### 3.3.1 Exercises
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#### 3.4 Tails of Distributions
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##### 3.4.1 Classification Based on Moments
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##### 3.4.2 Comparison Based on Limiting Tail
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##### 3.4.3 Classification Based on the Hazard Rate Function
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##### 3.4.4 Classification Based on the Mean Excess Loss Function The mean excess
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##### 3.4.5 Equilibrium Distributions and Tail Behavior
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##### 3.4.6 Exercises
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#### 3.5 Measures of Risk
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##### 3.5.1 Introduction
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##### 3.5.2 Risk Measures and Coherence
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##### 3.5.3 Value at Risk
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##### 3.5.4 Tail Value of Risk
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##### 3.5.5 Exercises
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## Part II Actuarial Models
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### 4. Characteristics Of Actuarial Models
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#### 4.1 Introduction
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#### 4.2 The Role of Parameters
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##### 4.2.1 Parametric and Scale Distributions
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##### 4.2.2 Parametric Distribution Families
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##### 4.2.3 Finite Mixture Distributions
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##### 4.2.4 Data-Dependent Distributions
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##### 4.2.5 Exercises
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### 5. Continuous Models
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#### 5.1 Introduction
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#### 5.2 Creating New Distributions
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##### 5.2.1 Multiplication by a Constant
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##### 5.2.2 Raising to a Power
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##### 5.2.3 Exponentiation
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##### 5.2.4 Mixing
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##### 5.2.5 Frailty Models
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##### 5.2.6 Splicing
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##### 5.2.7 Exercises
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#### 5.3 Selected Distributions and Their Relationships
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##### 5.3.1 Introduction
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##### 5.3.2 Two Parametric Families
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##### 5.3.3 Limiting Distributions
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##### 5.3.4 Two Heavy-Tailed Distributions
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##### 5.3.5 Exercises
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#### 5.4 The Linear Exponential Family
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##### 5.4.1 Exercises
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### 6. Discrete Distributions
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#### 6.1 Introduction
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##### 6.1.1 Exercise
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#### 6.2 The Poisson Distribution
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#### 6.3 The Negative Binomial Distribution
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#### 6.4 The Binomial Distribution
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#### 6.5 The (a,b) Class
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##### 6.5.1 Exercises
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#### 6.6 Truncation and Modification at zero
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##### 6.6.1 Exercises
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### 7. Advanced Discrete Distributions
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#### 7.1 Compound Frequency Distributions
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##### 7.1.1 Exercises
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#### 7.2 Further Properties of the Compound Poisson Class
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##### 7.2.1 Exercises
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#### 7.3 Mixed-Frequency Distributions
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##### 7.3.1 The General Mixed-Frequency Distribution
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##### 7.3.2 Mixed Poisson Distributions
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##### 7.3.3 Exercises
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#### 7.4 The Effect of Exposure on Frequency
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#### 7.5 An Inventory of Discrete Distributions
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##### 7.5.1 Exercises
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### 8. Frequency And Severity With Coverage Modifications
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#### 8.1 Introduction
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#### 8.2 Deductibles
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##### 8.2.1 Exercises
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#### 8.3 The Loss Elimination Ratio and the Effect of Inflation for Ordinary Deductibles
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##### 8.3.1 Exercise
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#### 8.4 Policy Limits
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##### 8.4.1 Exercises
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#### 8.5 Coinsurance, Deductibles, and Limits
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##### 8.5.1 Exercises
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#### 8.6 The Impact of Deductibles on Claim Frequency
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##### 8.6.1 Exercises
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### 9. Aggregate Loss Models
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#### 9.1 Introduction
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##### 9.1.1 Exercises
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#### 9.2 Model Choices
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##### 9.2.1 Exercises
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#### 9.3 The Compound Model for Aggregate Claims
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##### 9.3.1 Probabilities and Moments
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##### 9.3.2 Stop-Loss Insurance
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##### 9.3.3 The Tweedle Distribution
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##### 9.3.4 Exercises
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#### 9.4 Analytic Results
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##### 9.4.1 Exercises
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#### 9.5 Computing the Aggregate Claims Distribution
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#### 9.6 The Recursive Method
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##### 9.6.1 Applications to Compound Frequency Models
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##### 9.6.2 Underflow/Overflow Problems
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##### 9.6.3 Numerical Stability
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##### 9.6.4 Continuous Severity
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##### 9.6.5 Constructing Arithmetic Distributions
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##### 9.6.6 Exercises
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#### 9.7 The Impact of Individual Policy Modifications on Aggregate Payments
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##### 9.7.1 Exercises
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#### 9.8 The individual Risk Model
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##### 9.8.1 The Model
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##### 9.8.2 Parametric Approximation
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##### 9.8.3 Compound Poisson Approximation
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##### 9.8.4 Exercises
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## Part III Mathematical Statistics
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### 10. Introduction To Mathematical Statistics
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#### 10.1 Introduction and Four Data Sets
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#### 10.2 Point Estimation
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##### 10.2.1 Introduction
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##### 10.2.2 Measures of Quality
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##### 10.2.3 Exercises
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#### 10.3 Interval Estimation
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##### 10.3.1 Exercises
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#### 10.4 The Construction of Parametric Estimators
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##### 10.4.1 The Method of Moments and Percentile Matching
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##### 10.4.2 Exercises
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#### 10.5 Tests of Hypotheses
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##### 10.5.1 Exercise
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### 11. Maximum Likelihood Estimation
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#### 11.1 Introduction
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#### 11.2 Individual Data
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##### 11.2.1 Exercises
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#### 11.3 Grouped Data
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##### 11.3.1 Exercises
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#### 11.4 Truncated or Censored Data
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##### 11.4.1 Exercises
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#### 11.5 Variance and Interval Estimation for Maximum Likelihood Estimators
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##### 11.5.1 Exercises
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#### 11.6 Functions of Asymptotically Normal Estimators
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##### 11.6.1 Exercises
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#### 11.7 Nonnormal Confidence Intervals
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##### 11.7.1 Exercise
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### 12. Frequentist Estimation For Discrete Distributions
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#### 12.1 The Poisson Distribution
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#### 12.2 The Negative Binomial Distribution
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#### 12.3 The Binomial Distribution
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#### 12.4 The (a,b) Class
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#### 12.5 Compound Models
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#### 12.6 The Effect of Exposure on Maximum Likelihood Estimation
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#### 12.7 Exercises
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### 13. Bayesian Estimation
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#### 13.1 Definitions and Bayes' Theorem
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#### 13.2 Inference and Prediction
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##### 13.2.1 Exercises
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#### 13.3 Conjugate Prior Distributions and the Linear Exponential Family
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##### 13.3.1 Exercises
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#### 13.4 Computational Issues
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## Part IV Construction Of Models
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### 14. Construction Of Empirical Models
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#### 14.1 The Empirical Distribution
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#### 14.2 Empirical Distributions for Grouped Data
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##### 14.2.1 Exercises
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#### 14.3 Empirical Estimation with Right Censored Data
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##### 14.3.1 Exercises
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#### 14.4 Empirical Estimation of Moments
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##### 14.4.1 Exercises
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#### 14.5 Empirical Estimation with Left Truncated Data
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##### 14.5.1 Exercises
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#### 14.6 Kernel Density Models
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##### 14.6.1 Exercises
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#### 14.7 Approximations for large Data Sets
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##### 14.7.1 Introduction
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##### 14.7.2 Using Individual Data Points
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##### 14.7.3 Interval-Based Methods
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##### 14.7.4 Exercises
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#### 14.8 Maximum Likelihood Estimation of Decrement Probabilities
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##### 14.8.1 Exercise
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#### 14.9 Estimation of Transition Intensities
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### 15. Model Selection
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#### 15.1 Introduction
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#### 15.2 Representations of the Data and Model
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#### 15.3 Graphical Comparison of the Density and Distribution Functions
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##### 15.3.1 Exercises
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#### 15.4 Hypothesis Tests
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##### 15.4.1 The Kolmogorov--Smirnov Test
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##### 15.4.2 The Anderson--Darling Test
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##### 15.4.3 The Chi-Square Goodness-of-Fit Test
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##### 15.4.4 The Likelihood Ratio Test
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##### 15.4.5 Exercises
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#### 15.5 Selecting a Model
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##### 15.5.1 Introduction
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##### 15.2.2 Judgement-Based Approaches
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##### 15.5.3 Score-Based Approaches
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##### 15.5.4 Exercises
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## Part V Credibility
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### 16. Introduction To Limited Fluctuation Credibility
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#### 16.1 Introduction
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#### 16.2 Limited Fluctuation Credibility Theory
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#### 16.3 Full Credibility
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#### 16.4 Partial Credibility
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#### 16.5 Problems with the Approach
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#### 16.6 Notes and References
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#### 16.7 Exercises
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### 17. Greatest Accuracy Credibility
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#### 17.1 Introduction
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#### 17.2 Conditional Distributions and Expectation
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#### 17.3 The Bayesian Methodology
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#### 17.4 The Credibility Premium
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#### 17.5 The Bühlmann Model
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#### 17.6 The Bühlmann--Straub Model
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#### 17.7 Exact Credibility
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#### 17.8 Notes and References
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#### 17.9 Exercises
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### 18. Empirical Bayes Parameter Estimation
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#### 18.1 Introduction
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#### 18.2 Nonparametric Estimation
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#### 18.3 Semiparametric Estimation
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#### 18.4 Notes and References
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#### 18.5 Exercises
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## Part VI Simulation
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### 19. Simulation
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#### 19.1 Basics of Simulation
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##### 19.1.1 The Simulation Approach
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##### 19.1.2 Exercises
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#### 19.2 Simulation for Specific Distributions
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##### 19.2.1 Discrete Mixtures
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##### 19.2.2 Time or Age of Death from a Life Table
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##### 19.2.3 Simulating from the (a,b) Class
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##### 19.2.4 Normal and Lognormal Distributions
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##### 19.2.5 Exercises
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#### 19.3 Determining the Sample Size
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##### 19.3.1 Exercises
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#### 19.4 Examples of Simulation in Actuarial Modeling
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##### 19.4.1 Aggregate Loss Calculations
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##### 19.4.2 Examples of Lack of Independence
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##### 19.4.3 Simulation Analysis of the Two Examples
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##### 19.4.4 The Use of Simulation to Determine Risk Measures
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##### 19.4.5 Statistical Analyses
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##### 19.4.6 Exercises
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## Appendix A An Inventory Of Continuous Distributions
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### A.1 Introduction
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### A.2 The Transformed Beta Family
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#### A.2.1 The Four-Parameter Distribution
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#### A.2.2 Three-Parameter Distributions
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#### A.2.3 Two-Parameter Distributions
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### A.3 The Transformed Gamma Family
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#### A.3.1 Three-Parameter Distributions
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#### A.3.2 Two-Parameter Distributions
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#### A.3.3 One-Parameter Distributions
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### A.4 Distributions For Large Losses
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#### A.4.1 Extreme Value Distributions
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#### A.4.2 Generalized Pareto Distributions
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### A.4.5 Other Distributions
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### A.4.6 Distributions With Finite Support
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## Appendix B An Inventory Of Discrete Distributions
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### B.1 Introduction
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### B.2 The (a,b,0) Class
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### B.3 The (a,b,1) Class
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#### B.3.1 The Zero-Truncated Subclass
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#### B.3.2 The Zero-Modified Subclass
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### B.4 The Compound Class
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#### B.4.1 Some Compound Distributions
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### B.5 A Hierarchy Of Discrete Distributions
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## Appendix C Frequency And Severity Relationships
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## Appendix D The Recursive Formula
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## Appendix E Discretization Of The Severity Distribution
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### E.1 The Method Of Rounding
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### E.2 Mean Preserving
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### E.3 Undiscretization Of A Discretized Distribution
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## References
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## Index
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