https://github.com/openacttexts/Loss-Data-Analytics" />
Loss Data Analytics
Preface
Acknowledgements
Contributors
Reviewers
For our Readers
1
Pengenalan kepada Analisis Data Rugi
1.1
Relevan Analitis
1.1.1
What is Analytics?
1.1.2
Short and Long-term Insurance
1.1.3
Insurance Processes
1.2
Insurance Company Operations
1.2.1
Initiating Insurance
1.2.2
Renewing Insurance
1.2.3
Claims and Product Management
1.2.4
Loss Reserving
1.3
Case Study: Wisconsin Property Fund
1.3.1
Fund Claims Variables: Frequency and Severity
1.3.2
Fund Rating Variables
1.3.3
Fund Operations
1.4
Further Resources and Contributors
2
Frequency Modeling
2.1
Frequency Distributions
2.1.1
How Frequency Augments Severity Information
2.2
Basic Frequency Distributions
2.2.1
Foundations
2.2.2
Moment and Probability Generating Functions
2.2.3
Important Frequency Distributions
2.3
The (a, b, 0) Class
2.4
Estimating Frequency Distributions
2.4.1
Parameter estimation
2.4.2
Frequency Distributions MLE
2.5
Other Frequency Distributions
2.5.1
Zero Truncation or Modification
2.6
Mixture Distributions
2.7
Goodness of Fit
2.8
Exercises
2.9
R Code for Plots in this Chapter
2.10
Further Resources and Contributors
3
Modeling Loss Severity
3.1
Basic Distributional Quantities
3.1.1
Moments
3.1.2
Quantiles
3.1.3
Moment Generating Function
3.1.4
Probability Generating Function
3.2
Continuous Distributions for Modeling Loss Severity
3.2.1
Gamma Distribution
3.2.2
Pareto Distribution
3.2.3
Weibull Distribution
3.2.4
The Generalized Beta Distribution of the Second Kind
3.3
Methods of Creating New Distributions
3.3.1
Functions of Random Variables and their Distributions
3.3.2
Multiplication by a Constant
3.3.3
Raising to a Power
3.3.4
Exponentiation
3.3.5
Finite Mixtures
3.3.6
Continuous Mixtures
3.4
Coverage Modifications
3.4.1
Policy Deductibles
3.4.2
Policy Limits
3.4.3
Coinsurance
3.4.4
Reinsurance
3.5
Maximum Likelihood Estimation
3.5.1
Maximum Likelihood Estimators for Complete Data
3.5.2
Maximum Likelihood Estimators for Grouped Data
3.5.3
Maximum Likelihood Estimators for Censored Data
3.5.4
Maximum Likelihood Estimators for Truncated Data
3.6
Further Resources and Contributors
4
Model Selection and Estimation
4.1
Nonparametric Inference
4.1.1
Nonparametric Estimation
4.1.2
Tools for Model Selection and Diagnostics
4.1.3
Starting Values
4.2
Model Selection
4.2.1
Iterative Model Selection
4.2.2
Model Selection Based on a Training Dataset
4.2.3
Model Selection Based on a Test Dataset
4.2.4
Model Selection Based on Cross-Validation
4.3
Estimation using Modified Data
4.3.1
Parametric Estimation using Modified Data
4.3.2
Nonparametric Estimation using Modified Data
4.4
Bayesian Inference
4.4.1
Introduction to Bayesian Inference
4.4.2
Bayesian Model
4.4.3
Bayesian Inference
4.4.4
Conjugate Distributions
4.5
Further Resources and Contributors
Technical Supplement A. Gini Statistic
TS A.1. The Classic Lorenz Curve
TS A.2. Ordered Lorenz Curve and the Gini Index
TS A.3. Out-of-Sample Validation
5
Aggregate Loss Models
5.1
Introduction
5.2
Individual Risk Model
5.3
Collective Risk Model
5.3.1
Moments and Distribution
5.3.2
Stop-loss Insurance
5.3.3
Analytic Results
5.3.4
Tweedie Distribution
5.4
Computing the Aggregate Claims Distribution
5.4.1
Recursive Method
5.4.2
Simulation
5.5
Effects of Coverage Modifications
5.5.1
Impact of Exposure on Frequency
5.5.2
Impact of Deductibles on Claim Frequency
5.5.3
Impact of Policy Modifications on Aggregate Claims
5.6
Further Resources and Contributors
Technical Supplement B. Aggregate Loss Models
TS B.1. Individual Risk Model Properties
TS B.2. Relationship Between Probability Generating Functions of
\(X_i\)
and
\(X_i^T\)
TS B.3. Example 5.3.8 Moment Generating Function of Aggregate Loss
\(S_N\)
6
Simulation
6.1
Generating Independent Uniform Observations
6.2
Inverse Transform
6.3
How Many Simulated Values?
7
Premium Calculation Fundamentals
8
Risk Classification
8.1
Introduction
8.2
Poisson Regression Model
8.2.1
Need for Poisson Regression
8.2.2
Poisson Regression
8.2.3
Incorporating Exposure
8.2.4
Exercises
8.3
Categorical Variables and Multiplicative Tariff
8.3.1
Rating Factors and Tariff
8.3.2
Multiplicative Tariff Model
8.3.3
Poisson Regression for Multiplicative Tariff
8.3.4
Numerical Examples
8.4
Contributors and Further Resources
8.5
Technical Supplement – Estimating Poisson Regression Models
9
Experience Rating Using Credibility Theory
9.1
Introduction to Applications of Credibility Theory
9.2
Limited Fluctuation Credibility
9.2.1
Full Credibility for Claim Frequency
9.2.2
Full Credibility for Aggregate Losses and Pure Premium
9.2.3
Full Credibility for Severity
9.2.4
Partial Credibility
9.3
Bühlmann Credibility
9.3.1
Credibility Z,
EPV
, and
VHM
9.4
Bühlmann-Straub Credibility
9.5
Bayesian Inference and Bühlmann
9.5.1
Gamma-Poisson Model
9.5.2
Exact Credibility
9.6
Estimating Credibility Parameters
9.6.1
Full Credibility Standard for Limited Fluctuation Credibility
9.6.2
Nonparametric Estimation for Bühlmann and Bühlmann-Straub Models
9.6.3
Semiparametric Estimation for Bühlmann and Bühlmann-Straub Models
9.6.4
Balancing Credibility Estimators
9.7
Further Resources and Contributors
10
Insurance Portfolio Management including Reinsurance
Overview
10.1
Tails of Distributions
10.1.1
Classification Based on Moments
10.1.2
Comparison Based on Limiting Tail Behavior
10.2
Risk Measures
10.2.1
Coherent Risk Measures
10.2.2
Value-at-Risk
10.2.3
Tail Value-at-Risk
10.3
Reinsurance
10.3.1
Proportional Reinsurance
10.3.2
Non-Proportional Reinsurance
10.3.3
Additional Reinsurance Treaties
11
Loss Reserving
12
Experience Rating using Bonus-Malus
13
Data Systems
13.1
Data
13.1.1
Data Types and Sources
13.1.2
Data Structures and Storage
13.1.3
Data Quality
13.1.4
Data Cleaning
13.2
Data Analysis Preliminary
13.2.1
Data Analysis Process
13.2.2
Exploratory versus Confirmatory
13.2.3
Supervised versus Unsupervised
13.2.4
Parametric versus Nonparametric
13.2.5
Explanation versus Prediction
13.2.6
Data Modeling versus Algorithmic Modeling
13.2.7
Big Data Analysis
13.2.8
Reproducible Analysis
13.2.9
Ethical Issues
13.3
Data Analysis Techniques
13.3.1
Exploratory Techniques
13.3.2
Descriptive Statistics
13.3.3
Cluster Analysis
13.3.4
Confirmatory Techniques
13.4
Some R Functions
13.5
Summary
13.6
Further Resources and Contributors
14
Dependence Modeling
14.1
Variable Types
14.1.1
Qualitative Variables
14.1.2
Quantitative Variables
14.1.3
Multivariate Variables
14.2
Classic Measures of Scalar Associations
14.2.1
Association Measures for Quantitative Variables
14.2.2
Rank Based Measures
14.2.3
Nominal Variables
14.3
Introduction to Copulas
14.4
Application Using Copulas
14.4.1
Data Description
14.4.2
Marginal Models
14.4.3
Probability Integral Transformation
14.4.4
Joint Modeling with Copula Function
14.5
Types of Copulas
14.5.1
Elliptical Copulas
14.5.2
Archimedian Copulas
14.5.3
Properties of Copulas
14.6
Why is Dependence Modeling Important?
14.7
Further Resources and Contributors
Technical Supplement A. Other Classic Measures of Scalar Associations
A.1. Blomqvist’s Beta
A.2. Nonparametric Approach Using Spearman Correlation with Tied Ranks
15
Appendix A: Review of Statistical Inference
15.1
Basic Concepts
15.1.1
Random Sampling
15.1.2
Sampling Distribution
15.1.3
Central Limit Theorem
15.2
Point Estimation and Properties
15.2.1
Method of Moments Estimation
15.2.2
Maximum Likelihood Estimation
15.3
Interval Estimation
15.3.1
Exact Distribution for Normal Sample Mean
15.3.2
Large-sample Properties of MLE
15.3.3
Confidence Interval
15.4
Hypothesis Testing
15.4.1
Basic Concepts
15.4.2
Student-
\(t\)
test based on MLE
15.4.3
Likelihood Ratio Test
15.4.4
Information Criteria
16
Appendix B: Iterated Expectations
16.1
Conditional Distribution and Conditional Expectation
16.1.1
Conditional Distribution
16.1.2
Conditional Expectation and Conditional Variance
16.2
Iterated Expectations and Total Variance
16.2.1
Law of Iterated Expectations
16.2.2
Law of Total Variance
16.2.3
Application
16.3
Conjugate Distributions
16.3.1
Linear Exponential Family
16.3.2
Conjugate Distributions
17
Appendix C: Maximum Likelihood Theory
17.1
Likelihood Function
17.1.1
Likelihood and Log-likelihood Functions
17.1.2
Properties of Likelihood Functions
17.2
Maximum Likelihood Estimators
17.2.1
Definition and Derivation of MLE
17.2.2
Asymptotic Properties of MLE
17.2.3
Use of Maximum Likelihood Estimation
17.3
Statistical Inference Based on Maximum Likelhood Estimation
17.3.1
Hypothesis Testing
17.3.2
MLE and Model Validation
Bibliography
Loss Data Analytics on GitHub
Loss Data Analytics
Chapter 7
Premium Calculation Fundamentals
This is a placeholder file