Computational Actuarial Science with R

Computational Actuarial Science with R

by Arthur Charpentier
Released August 2014
Publisher(s): Chapman and Hall/CRC
ISBN: 9781498759823

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Book description

A Hands-On Approach to Understanding and Using Actuarial ModelsComputational Actuarial Science with R provides an introduction to the computational aspects of actuarial science. Using simple R code, the book helps you understand the algorithms involved in actuarial computations. It also covers more advanced topics, such as parallel computing and C/

Table of contents

  1. Preliminaries
  2. Preface
  3. Contributors
  4. Chapter 1 Introduction
    1. 1.1 R for Actuarial Science?
      1. 1.1.1 From Actuarial Science to Computational Actuarial Science
      2. 1.1.2 The S Language and the R Environment
      3. 1.1.3 Vectors and Matrices in Actuarial Computations
      4. 1.1.4 R Packages
      5. 1.1.5 S3 versus S4 Classes
      6. 1.1.6 R Codes and Efficiency
    2. 1.2 Importing and Creating Various Objects, and Datasets in R
      1. 1.2.1 Simple Objects in R and Workspace
      2. 1.2.2 More Complex Objects in R: From Vectors to Lists
        1. 1.2.2.1 Vectors in R
        2. 1.2.2.2 Matrices and Arrays
        3. 1.2.2.3 Lists
      3. 1.2.3 Reading csv or txt Files
      4. 1.2.4 Importing Excel® Files and SAS® Tables
      5. 1.2.5 Characters, Factors and Dates with R
        1. 1.2.5.1 Strings and Characters
        2. 1.2.5.2 Factors and Categorical Variables
        3. 1.2.5.3 Dates in R
      6. 1.2.6 Symbolic Expressions in R
    3. 1.3 Basics of the R Language
      1. 1.3.1 Core Functions
      2. 1.3.2 From Control Flow to “Personal” Functions
        1. 1.3.2.1 Control Flow: Looping, Repeating and Conditioning
        2. 1.3.2.2 Writing Personal Functions
      3. 1.3.3 Playing with Functions (in a Life Insurance Context)
      4. 1.3.4 Dealing with Errors
      5. 1.3.5 Efficient Functions
      6. 1.3.6 Numerical Integration
      7. 1.3.7 Graphics with R: A Short Introduction
        1. 1.3.7.1 Basic Ready-Made Graphs
        2. 1.3.7.2 A Simple Graph with Lines and Curves
        3. 1.3.7.3 Graphs That Can Be Obtained from Standard Functions
        4. 1.3.7.4 Adding Shaded Area to a Graph
        5. 1.3.7.5 3D Graphs
        6. 1.3.7.6 More Complex Graphs
    4. 1.4 More Advanced R
      1. 1.4.1 Memory Issues
      2. 1.4.2 Parallel R
      3. 1.4.3 Interfacing R and C/C++
      4. 1.4.4 Integrating R in Excel®
      5. 1.4.5 Going Further
    5. 1.5 Ending an R Session
    6. 1.6 Exercises
      1. Figure 1.1
      2. Figure 1.2
      3. Figure 1.3
      4. Figure 1.4
      5. Figure 1.5
      6. Figure 1.6
      7. Figure 1.7
      8. Figure 1.8
      1. Table 1.1
      2. Table 1.2
  5. Part I Methodology
  6. Chapter 2 Standard Statistical Inference
    1. 2.1 Probability Distributions in Actuarial Science
      1. 2.1.1 Continuous Distributions
      2. 2.1.2 Discrete Distributions
      3. 2.1.3 Mixed-Type Distributions
      4. 2.1.4 S3 versus S4 Types for Distribution
    2. 2.2 Parametric Inference
      1. 2.2.1 Maximum Likelihood Estimation
      2. 2.2.2 Moment Matching Estimation
      3. 2.2.3 Quantile Matching Estimation
      4. 2.2.4 Maximum Goodness-of-Fit Estimation
    3. 2.3 Measures of Adequacy
      1. 2.3.1 Histogram and Empirical Densities
      2. 2.3.2 Distribution Function Plot
      3. 2.3.3 QQ-Plot, PP-Plot
      4. 2.3.4 Goodness-of-Fit Statistics and Tests
      5. 2.3.5 Skewness—Kurtosis Graph
    4. 2.4 Linear Regression: Introducing Covariates in Statistical Infer­ence
      1. 2.4.1 Using Covariates in the Statistical Framework
      2. 2.4.2 Linear Regression Model
      3. 2.4.3 Inference in a Linear Model
    5. 2.5 Aggregate Loss Distribution
      1. 2.5.1 Computation of the Aggregate Loss Distribution
      2. 2.5.2 Poisson Process
      3. 2.5.3 From Poisson Processes to Levy Processes
      4. 2.5.4 Ruin Models
    6. 2.6 Copulas and Multivariate Distributions
      1. 2.6.1 Definition of Copulas
      2. 2.6.2 Archimedean Copulas
      3. 2.6.3 Elliptical Copulas
      4. 2.6.4 Properties and Extreme Copulas
      5. 2.6.5 Copula Fitting Methods
      6. 2.6.6 Application and Copula Selection
    7. 2.7 Exercises
      1. Figure 2.1
      2. Figure 2.2
      3. Figure 2.3
      4. Figure 2.4
      5. Figure 2.5
      6. Figure 2.6
      7. Figure 2.7
      8. Figure 2.8
      9. Figure 2.9
      10. Figure 2.10
      11. Figure 2.11
      12. Figure 2.12
      13. Figure 2.13
      14. Figure 2.14
      15. Figure 2.15
      16. Figure 2.16
      17. Figure 2.17
      18. Figure 2.18
      19. Figure 2.19
      20. Figure 2.20
      21. Figure 2.21
      1. Table 2.1
  7. Chapter 3 Bayesian Philosophy
    1. 3.1 Introduction
      1. 3.1.1 A Formal Introduction
      2. 3.1.2 Two Kinds of Probability
      3. 3.1.3 Working with Subjective Probabilities in Real Life
      4. 3.1.4 Bayesianism for Actuaries
    2. 3.2 Bayesian Conjugates
      1. 3.2.1 Historical Perspective
      2. 3.2.2 Motivation on Small Samples
      3. 3.2.3 Black Swans and Bayesian Methodology
      4. 3.2.4 Bayesian Models in Portfolio Management and Finance
      5. 3.2.5 Relation to Biihlmann Credibility
      6. 3.2.6 Noninformative Priors
    3. 3.3 Computational Considerations
      1. 3.3.1 Curse of Dimensionality
      2. 3.3.2 Monte Carlo Integration
      3. 3.3.3 Markov Chain Monte Carlo
      4. 3.3.4 MCMC Example in R
      5. 3.3.5 JAGS and Stan
      6. 3.3.6 Computational Conclusion and Specific Packages
    4. 3.4 Bayesian Regression
      1. 3.4.1 Linear Model from a Bayesian Perspective
      2. 3.4.2 Extension to Generalized Linear Models
      3. 3.4.3 Extension for Hierarchical Structures
    5. 3.5 Interpretation of Bayesianism
      1. 3.5.1 Bayesianism and Decision Theory
      2. 3.5.2 Context of Discovery versus Context of Justification
      3. 3.5.3 Practical Classical versus Bayesian Statistics Revisited
    6. 3.6 Conclusion
    7. 3.7 Exercises
      1. Figure 3.1
      2. Figure 3.2
      3. Figure 3.3
      4. Figure 3.4
      5. Figure 3.5
      6. Figure 3.6
      7. Figure 3.7
      1. Table 3.1
      2. Table 3.2
  8. Chapter 4 Statistical Learning
    1. 4.1 Introduction and Motivation
      1. 4.1.1 The Dataset
      2. 4.1.2 Description of the Data
      3. 4.1.3 Scoring Tools
      4. 4.1.4 Recoding the Variables
      5. 4.1.5 Training and Testing Samples
    2. 4.2 Logistic Regression
      1. 4.2.1 Inference in the Logistic Model
      2. 4.2.2 Logistic Regression on Categorical Variates
      3. 4.2.3 Step-by-Step Variable Selection
        1. 4.2.3.1 Forward Algorithm
        2. 4.2.3.2 Backward Algorithm
      4. 4.2.4 Leaps and Bounds
      5. 4.2.5 Smoothing Continuous Covariates
      6. 4.2.6 Nearest-Neighbor Method
    3. 4.3 Penalized Logistic Regression: From Ridge to Lasso
      1. 4.3.1 Ridge Model
      2. 4.3.2 Lasso Regression
    4. 4.4 Classification and Regression Trees
      1. 4.4.1 Partitioning
      2. 4.4.2 Criteria and Impurity
    5. 4.5 From Classification Trees to Random Forests
      1. 4.5.1 Bagging
      2. 4.5.2 Boosting
      3. 4.5.3 Random Forests
      1. Figure 4.1
      2. Figure 4.2
      3. Figure 4.3
      4. Figure 4.4
      5. Figure 4.5
      6. Figure 4.6
      7. Figure 4.7
      8. Figure 4.8
      9. Figure 4.9
      10. Figure 4.10
      11. Figure 4.11
      12. Figure 4.12
      13. Figure 4.14
      14. Figure 4.13
      15. Figure 4.15
      16. Figure 4.16
      17. Figure 4.18
      18. Figure 4.17
      19. Figure 4.20
      20. Figure 4.19
      21. Figure 4.21
      22. Figure 4.22
      1. Table 4.1
      2. Table 4.2
  9. Chapter 5 Spatial Analysis
    1. 5.1 Introduction
      1. 5.1.1 Point Pattern Data
      2. 5.1.2 Random Surface Data
      3. 5.1.3 Spatial Interaction Data
      4. 5.1.4 Areal Data
      5. 5.1.5 Focus of This Chapter
    2. 5.2 Spatial Analysis and GIS
    3. 5.3 Spatial Objects in R
      1. 5.3.1 SpatialPoints Subclass
      2. 5.3.2 SpatialPointsDataFrame Subclass
      3. 5.3.3 SpatialPolygons Subclass
        1. 5.3.3.1 First Elementary Example
        2. 5.3.3.2 Second Example
      4. 5.3.4 SpatialPolygonsDataFrame Subclass
    4. 5.4 Maps in R
    5. 5.5 Reading Maps and Data in R
    6. 5.6 Exploratory Spatial Data Analysis
      1. 5.6.1 Mapping a Variable
      2. 5.6.2 Selecting Colors
      3. 5.6.3 Using the RgoogleMaps Package
      4. 5.6.4 Generating KML Files
        1. 5.6.4.1 Adding a Legend to a KML File
    7. 5.7 Testing for Spatial Correlation
      1. 5.7.1 Neighborhood Matrix
      2. 5.7.2 Other Neighborhood Options
      3. 5.7.3 Moran's I Index
    8. 5.8 Spatial Car Accident Insurance Analysis
    9. 5.9 Spatial Car Accident Insurance Shared Analysis
    10. 5.10 Conclusion
      1. Figure 5.1
      2. Figure 5.2
      3. Figure 5.3
      4. Figure 5.4
      5. Figure 5.5
      6. Figure 5.6
      7. Figure 5.7
      8. Figure 5.8
      9. Figure 5.9
      10. Figure 5.10
      11. Figure 5.11
      12. Figure 5.12
      13. Figure 5.13
      14. Figure 5.15
      15. Figure 5.14
      16. Figure 5.16
      17. Figure 5.17
      18. Figure 5.18
      19. Figure 5.19
      20. Figure 5.20
      21. Figure 5.21
      22. Figure 5.22
      23. Figure 5.23
      1. Table 5.1
  10. Chapter 6 Reinsurance and Extremal Events
    1. 6.1 Introduction
    2. 6.2 Univariate Extremes
      1. 6.2.1 Block Maxima
      2. 6.2.2 Exceedances above a Threshold
      3. 6.2.3 Point Process
    3. 6.3 Inference
      1. 6.3.1 Visualizing Tails
      2. 6.3.2 Estimation
        1. 6.3.2.1 Generalized Extreme Value Distribution
        2. 6.3.2.2 Poisson-Generalized Pareto Model
        3. 6.3.2.3 Point Process
        4. 6.3.2.4 Other Tail Index Estimates
      3. 6.3.3 Checking for the Asymptotic Regime Assumption
        1. 6.3.3.1 Mean Excess Plot
        2. 6.3.3.2 Parameter Stability
      4. 6.3.4 Quantile Estimation
    4. 6.4 Model Checking
      1. 6.4.1 Quantile Quantile Plot
      2. 6.4.2 Probability—Probability Plot
      3. 6.4.3 Return Level Plot
    5. 6.5 Reinsurance Pricing
      1. 6.5.1 Modeling Occurence and Frequency
      2. 6.5.2 Modeling Individual Losses
      1. Figure 6.1
      2. Figure 6.2
      3. Figure 6.3
      4. Figure 6.4
      5. Figure 6.5
      6. Figure 6.6
      7. Figure 6.7
      8. Figure 6.8
      9. Figure 6.9
      10. Figure 6.10
      11. Figure 6.11
      12. Figure 6.12
      13. Figure 6.13
  11. Part II Life Insurance
  12. Chapter 7 Life Contingencies
    1. 7.1 Introduction
    2. 7.2 Financial Mathematics Review
    3. 7.3 Working with Life Tables
    4. 7.4 Pricing Life Insurance
    5. 7.5 Reserving Life Insurances
    6. 7.6 More Advanced Topics
    7. 7.7 Health Insurance and Markov Chains
      1. 7.7.1 Markov Chain with R
      2. 7.7.2 Valuation of Cash Flows
      3. 7.7.3 APV of Benefits and Reserves
    8. 7.8 Exercises
      1. 7.8.1 Financial Mathematics
      2. 7.8.2 Demography
      3. 7.8.3 Pricing Life Insurance
      4. 7.8.4 Reserving Life Insurances
      5. 7.8.5 More Advanced Topics
      1. Figure 7.1
      2. Figure 7.2
      3. Figure 7.3
  13. Chapter 8 Prospective Life Tables
    1. 8.1 Introduction
    2. 8.2 Smoothing Mortality Data
      1. 8.2.1 Weighted Constrained Penalized Regression Splines
      2. 8.2.2 Two-Dimensional P-Splines
    3. 8.3 Lee—Carter and Related Forecasting Methods
      1. 8.3.1 Lee—Carter (LC) Method
      2. 8.3.2 Lee—Miller (LM) Method
      3. 8.3.3 Booth—Maindonald—Smith (BMS) Method
      4. 8.3.4 Hyndman—Ullah (HU) Method
      5. 8.3.5 Robust Hyndman-Ullah (HUrob) Method
      6. 8.3.6 Weighted Hyndman-Ullah (HUw) Method
    4. 8.4 Other Mortality Forecasting Methods
    5. 8.5 Coherent Mortality Forecasting
    6. 8.6 Life Table Forecasting
    7. 8.7 Life Insurance Products
    8. 8.8 Exercises
      1. Figure 8.1
      2. Figure 8.2
      3. Figure 8.3
      4. Figure 8.4
      5. Figure 8.5
      6. Figure 8.6
      7. Figure 8.7
      8. Figure 8.8
      9. Figure 8.9
      10. Figure 8.10
      11. Figure 8.11
      12. Figure 8.12
      13. Figure 8.13
      14. Figure 8.14
      15. Figure 8.15
      16. Figure 8.16
      1. Table 8.1
  14. Chapter 9 Prospective Mortality Tables and Portfolio Experience
    1. 9.1 Introduction and Motivation
    2. 9.2 Notation, Data, and Assumption
    3. 9.3 The Methods
      1. 9.3.1 Method 1: Approach Involving One Parameter with the SMR
      2. 9.3.2 Method 2: Approach Involving Two Parameters with a Semiparametric Relational Model
      3. 9.3.3 Method 3: Poisson GLM Including Interactions with Age and Calendar Year
      4. 9.3.4 Method 4: Nonparametric Smoothing and Application of the Im­provement Rates
      5. 9.3.5 Completion of the Tables: The Approach of Denuit and Goderniaux
    4. 9.4 Validation
      1. 9.4.1 First Level: Proximity between the Observations and the Model
      2. 9.4.2 Second Level: Regularity of the Fit
      3. 9.4.3 Third Level: Consistency and Plausibility of the Mortality Trends
    5. 9.5 Operational Framework
      1. 9.5.1 The Package ELT
      2. 9.5.2 Computation of the Observed Statistics and Importation of the Reference
      3. 9.5.3 Execution of the Methods
      4. 9.5.4 Process of Validation
      5. 9.5.5 Completion of the Tables
      1. Figure 9.1
      2. Figure 9.2
      3. Figure 9.3
      4. Figure 9.4
      5. Figure 9.5
      6. Figure 9.6
      7. Figure 9.7
      8. Figure 9.8
      9. Figure 9.9
      10. Figure 9.10
      1. Table 9.1
      2. Table 9.2
      3. Table 9.3
      4. Table 9.4
  15. Chapter 10 Survival Analysis
    1. 10.1 Introduction
    2. 10.2 Working with Incomplete Data
      1. 10.2.1 Data Importation and Some Statistics
      2. 10.2.2 Building the Appropriate Database
      3. 10.2.3 Some Descriptive Statistics
    3. 10.3 Survival Distribution Estimation
      1. 10.3.1 Hoem Estimator of the Conditional Rates
      2. 10.3.2 Kaplan—Meier Estimator of the Survival Function
    4. 10.4 Regularization Techniques
      1. 10.4.1 Parametric Adjustment
      2. 10.4.2 Semiparametric Adjustment: Brass Relational Model
      3. 10.4.3 Nonparametric Techniques: Whittaker—Henderson Smoother
      4. 10.4.3.1 Application
    5. 10.5 Modeling Heterogeneity
      1. 10.5.1 Semiparametric Framework: Cox Model
      2. 10.5.2 Additive Models
    6. 10.6 Validation of a Survival Model
      1. Figure 10.1
      2. Figure 10.2
      3. Figure 10.3
      4. Figure 10.4
      5. Figure 10.5
      6. Figure 10.6
      7. Figure 10.7
      8. Figure 10.8
  16. Part III Finance
  17. Chapter 11 Stock Prices and Time Series
    1. 11.1 Introduction
    2. 11.2 Financial Time Series
      1. 11.2.1 Introduction
      2. 11.2.2 Data Used in This Chapter
      3. 11.2.3 Stylized Facts
    3. 11.3 Heteroskedastic Models
      1. 11.3.1 Introduction
      2. 11.3.2 Standard GARCH(1,1) Model
      3. 11.3.3 Forecasting Heteroskedastic Model
      4. 11.3.4 More Efficient Implementation
    4. 11.4 Application: Estimation of the VaR Based on the POT and GARCH Model
    5. 11.5 Conclusion
      1. Figure 11.1
      2. Figure 11.2
      3. Figure 11.3
      1. Table 11.1
  18. Chapter 12 Yield Curves and Interest Rates Models
    1. 12.1 A Brief Overview of the Yield Curve and Scenario Simulation
    2. 12.2 Yield Curves
      1. 12.2.1 Description of the Datasets
      2. 12.2.2 Principal Component Analysis
    3. 12.3 Nelson—Siegel Model
      1. 12.3.1 Estimating the Nelson-Siegel Model with R
    4. 12.4 Svensson Model
      1. 12.4.1 Estimating the Svensson Model with R
      1. Figure 12.2
      2. Figure 12.1
      3. Figure 12.3
      4. Figure 12.4
      5. Figure 12.5
      6. Figure 12.6
      7. Figure 12.7
      8. Figure 12.8
  19. Chapter 13 Portfolio Allocation
    1. 13.1 Introduction
    2. 13.2 Optimization Problems in R
      1. 13.2.1 Introduction
      2. 13.2.2 Linear Programming
      3. 13.2.3 Quadratic Programming
      4. 13.2.4 Nonlinear Programming
    3. 13.3 Data Sources
    4. 13.4 Portfolio Returns and Cumulative Performance
    5. 13.5 Portfolio Optimization in R
      1. 13.5.1 Introduction
      2. 13.5.2 Mean—Variance Portfolio
      3. 13.5.3 Robust Mean-Variance Portfolio
      4. 13.5.4 Minimum Variance Portfolio
      5. 13.5.5 Conditional Value-at-Risk Portfolio
      6. 13.5.6 Minimum Drawdown Portfolio
    6. 13.6 Display Results
      1. 13.6.1 Efficient Frontier
      2. 13.6.2 Weighted Return Plots
    7. 13.7 Conclusion
      1. Figure 13.1
      2. Figure 13.2
      3. Figure 13.3
      4. Figure 13.4
      5. Figure 13.5
      6. Figure 13.6
      7. Figure 13.7
      8. Figure 13.8
      1. Table 13.1
  20. Part IV Non-Life Insurance
  21. Chapter 14 General Insurance Pricing
    1. 14.1 Introduction and Motivation
      1. 14.1.1 Collective Model in General Insurance
      2. 14.1.2 Pure Premium in a Heterogenous Context
      3. 14.1.3 Dataset
      4. 14.1.4 Structure of the Chapter and References
    2. 14.2 Claims Frequency and Log-Poisson Regression
      1. 14.2.1 Annualized Claims Frequency
      2. 14.2.2 Poisson Regression
      3. 14.2.3 Ratemaking with One Categorical Variable
      4. 14.2.4 Contingency Tables and Minimal Bias Techniques
      5. 14.2.5 Ratemaking with Continuous Variables
      6. 14.2.6 A Poisson Regression to Model Yearly Claim Frequency
    3. 14.3 From Poisson to Quasi-Poisson
      1. 14.3.1 NB1 Variance Form: Negative Binomial Type I
      2. 14.3.2 NB2 Variance Form: Negative Binomial Type II
      3. 14.3.3 Unstructured Variance Form
      4. 14.3.4 Nonparametric Variance Form
    4. 14.4 More Advanced Models for Counts
      1. 14.4.1 Negative Binomial Regression
      2. 14.4.2 Zero-Inflated Models
      3. 14.4.3 Hurdle Models
    5. 14.5 Individual Claims, Gamma, Log-Normal, and Other Regres­sions
      1. 14.5.1 Gamma Regression
      2. 14.5.2 The Log-Normal Model
      3. 14.5.3 Gamma versus Log-Normal Models
      4. 14.5.4 Inverse Gaussian Model
    6. 14.6 Large Claims and Ratemaking
      1. 14.6.1 Model with Two Kinds of Claims
      2. 14.6.2 More General Model
    7. 14.7 Modeling Compound Sum with Tweedie Regression
    8. 14.8 Exercises
      1. Figure 14.1
      2. Figure 14.2
      3. Figure 14.3
      4. Figure 14.4
      5. Figure 14.5
      6. Figure 14.6
      7. Figure 14.7
      8. Figure 14.8
      9. Figure 14.9
  22. Chapter 15 Longitudinal Data and Experience Rating
    1. 15.1 Motivation
      1. 15.1.1 A Priori Rating for Cross-Sectional Data
      2. 15.1.2 Experience Rating for Panel Data
      3. 15.1.3 From Panel to Multilevel Data
      4. 15.1.4 Structure of the Chapter
    2. 15.2 Linear Models for Longitudinal Data
      1. 15.2.1 Data
      2. 15.2.2 Fixed Effects Models
      3. 15.2.3 Models with Serial Correlation
      4. 15.2.4 Models with Random Effects
      5. 15.2.5 Prediction
    3. 15.3 Generalized Linear Models for Longitudinal Data
      1. 15.3.1 Specifying Generalized Linear Models with Random Effects
      2. 15.3.2 Case Study: Experience Rating with Bonus—Malus Scales in R
        1. 15.3.2.1 Bonus—Malus Scales
        2. 15.3.2.2 Transition Rules, Transition Probabilities and Stationary Distribution
        3. 15.3.2.3 Relativities
      1. Figure 15.1
      2. Figure 15.2
      1. Table 15.1
      2. Table 15.2
      3. Table 15.3
      4. Table 15.4
      5. Table 15.5
  23. Chapter 16 Claims Reserving and IBNR
    1. 16.1 Introduction
      1. 16.1.1 Motivation
      2. 16.1.2 Outline and Scope
    2. 16.2 Development Triangles
    3. 16.3 Deterministic Reserving Methods
      1. 16.3.1 Chain-Ladder Algorithm
      2. 16.3.2 Tail Factors
    4. 16.4 Stochastic Reserving Models
      1. 16.4.1 Chain-Ladder in the Context of Linear Regression
      2. 16.4.2 Mack Model
      3. 16.4.3 Poisson Regression Model for Incremental Claims
      4. 16.4.4 Bootstrap Chain-Ladder
      5. 16.4.5 Reserving Based on Log-Incremental Payments
    5. 16.5 Quantifying Reserve Risk
      1. 16.5.1 Ultimo Reserve Risk
      2. 16.5.2 One-Year Reserve Risk
    6. 16.6 Discussion
    7. 16.7 Exercises
      1. Figure 16.1
      2. Figure 16.2
      3. Figure 16.3
      4. Figure 16.4
      5. Figure 16.5
      6. Figure 16.6
      7. Figure 16.7
      8. Figure 16.8
      9. Figure 16.9
      10. Figure 16.10
      11. Figure 16.11
      12. Figure 16.12
      13. Figure 16.13
      14. Figure 16.14
      15. Figure 16.15
  24. Chapter 17 Bibliography

Product information

  • Title: Computational Actuarial Science with R
  • Author(s): Arthur Charpentier
  • Release date: August 2014
  • Publisher(s): Chapman and Hall/CRC
  • ISBN: 9781498759823