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Department of Applied Finance and Actuarial Studies

Table of Contents

1 Insurance Data
    1.1 Introduction
    1.2 Types of variables
    1.3 Data transformations
    1.4 Data exploration
    1.5 Grouping and runoff triangles
    1.6 Assessing distributions
    1.7 Data issues and biases
    1.8 Data sets used
    1.9 Outline of rest of book

2 Response distributions
    2.1 Discrete and continuous random variables
    2.2 Bernoulli
    2.3 Binomial
    2.4 Poisson
    2.5 Negative binomial
    2.6 Normal
    2.7 Chi–square and gamma
    2.8 Inverse Gaussian
    2.9 Overdispersion

3 Exponential family responses and estimation
    3.1 Exponential family
    3.2 The variance function
    3.3 Proof of the mean and variance expressions
    3.4 Standard distributions in the exponential family form
    3.5 Fitting probability functions to data

4 Linear modelling
    4.1 History and terminology of linear modelling
    4.2 What does “linear” in linear model mean?
    4.3 Simple linear modelling
    4.4 Multiple linear modelling
    4.5 The classical linear model
    4.6 Least squares properties under the classical linear model
    4.7 Weighted least squares
    4.8 Grouped and ungrouped data
    4.9 Transformations to normality and linearity
    4.10 Categorical explanatory variables
    4.11 Polynomial regression
    4.12 Banding continuous explanatory variables
    4.13 Interaction
    4.14 Collinearity
    4.15 Hypothesis testing
    4.16 Checks using the residuals
    4.17 Checking explanatory variable specifications
    4.18 Outliers
    4.19 Model selection

5 Generalized Linear Models
    5.1 The generalized linear model
    5.2 Steps in generalized linear modelling
    5.3 Links and canonical links
    5.4 Use of an offset
    5.5 Maximum likelihood estimation
    5.6 Condence intervals and prediction
    5.7 Assessing fits and the deviance
    5.8 Testing the signicance of explanatory variables
    5.9 Residuals
    5.10 Further diagnostic tools
    5.11 Model selection

6 Models for count data

    6.1 Poisson regression
    6.2 Poisson overdispersion and negative binomial regression
    6.3 Quasi–likelihood
    6.4 Counts and frequencies

7 Categorical responses

    7.1 Binary responses
    7.2 Logistic regression
    7.3 Application of logistic regression to vehicle insurance
    7.4 Correcting for exposure
    7.5 Grouped binary data
    7.6 Goodness of fit for logistic regression
    7.7 Categorical responses with more than two categories
    7.8 Ordinal responses
    7.9 Nominal responses

8 Continuous responses
    8.1 Gamma regression
    8.2 Inverse Gaussian regression
    8.3 Tweedie regression

9 Correlated data

    9.1 Random effects
    9.2 Specication of within–cluster correlation
    9.3 Generalized estimating equations

10 Extensions to the Generalized Linear Model
    10.1 Generalized Additive Models
    10.2 Double generalized linear models
    10.3 Generalized additive models for location, scale and shape
    10.4 Zero–adjusted inverse Gaussian regression
    10.5 A mean and dispersion model for total claim size

Appendix 1 Computer code and output
    A1.1 Poisson regression
    A1.2 Negative binomial regression
    A1.3 Quasi-likelihood regression
    A1.4 Logistic regression
    A1.5 Ordinal regression
    A1.6 Nominal regression
    A1.7 Gamma regression
    A1.8 Inverse Gaussian regression
    A1.9 Logistic regression GLMM
    A1.10 Logistic regression GEE
    A1.11 Logistic regression GAM
    A1.12 GAMLSS
    A1.13 Zero-adjusted inverse Gaussian regression