generalized linear model.pdf

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Cover

Generalized Linear Models for Insurance Data

Title

Contents

Preface

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

Exercises

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

Exercises

4 Linear modeling

4.1 History and terminology of linear modeling

4.2 What does “linear” in linear model mean?

4.3 Simple linear modeling

4.4 Multiple linear modeling

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 modeling

5.3 Links and canonical links

5.4 Offsets

5.5 Maximum likelihood estimation

5.6 Confidence intervals and prediction

5.7 Assessing fits and the deviance

5.8 Testing the significance of explanatory variables

5.9 Residuals

5.10 Further diagnostic tools

5.11 Model selection

Exercises

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

Exercises

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

Exercises

8 Continuous responses

8.1 Gamma regression

8.2 Inverse Gaussian regression

8.3 Tweedie regression

Exercises

9 Correlated data

9.1 Random effects

9.2 Specification of within-cluster correlation

9.3 Generalized estimating equations

Exercise

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

Exercises

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

Bibliography

Index

Generalized Linear Models for Insurance Data Actuaries should have the tools they need. Generalized linear models are used in the insurance industry to support critical decisions. Yet no text intro- duces GLMs in this context and addresses problems speciﬁc to insurance data. Until now. Practical and rigorous, this books treats GLMs, covers all standard exponen- tial family distributions, extends the methodology to correlated data structures, and discusses other techniques of interest and how they contrast with GLMs. The focus is on issues which are speciﬁc to insurance data and all techniques are illustrated on data sets relevant to insurance. Exercises and data-based practicals help readers to consolidate their skills, with solutions and data sets given on the companion website. Although the book is package-independent, SAS code and output examples feature in an appendix and on the website. In addition, R code and output for all examples are provided on the website. International Series on Actuarial Science Mark Davis, Imperial College London John Hylands, Standard Life John McCutcheon, Heriot-Watt University Ragnar Norberg, London School of Economics H. Panjer, Waterloo University Andrew Wilson, Watson Wyatt The International Series on Actuarial Science, published by Cambridge University Press in conjunction with the Institute of Actuaries and the Faculty of Actuaries, will contain textbooks for students taking courses in or related to actuarial science, as well as more advanced works designed for continuing pro- fessional development or for describing and synthesizing research. The series will be a vehicle for publishing books that reﬂect changes and developments in the curriculum, that encourage the introduction of courses on actuarial science in universities, and that show how actuarial science can be used in all areas where there is long-term ﬁnancial risk.

GENERALIZED LINEAR MODELS FOR INSURANCE DATA P I E T D E J O N G Department of Actuarial Studies, Macquarie University, Sydney G I L L I A N Z . H E L L E R Department of Statistics, Macquarie University, Sydney

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, S˜ao Paulo, Delhi C A M B R I D G E U N I V E R S I T Y P R E S S Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: http://www.afas.mq.edu.au/research/books/glms for insurance data c P. de Jong and G. Z. Heller 2008 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2008 Third printing 2009 Printed in the United Kingdom at the University Press, Cambridge A catalog record for this publication is available from the British Library ISBN 978-0-521-87914-9 hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Information regarding prices, travel timetables and other factual information given in this work are correct at the time of ﬁrst printing but Cambridge University Press does not guarantee the accuracy of such information thereafter.

Contents Preface 1 2 3 Introduction Types of variables Data transformations Data exploration Grouping and runoff triangles Assessing distributions Data issues and biases Data sets used Outline of rest of book Insurance data 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 Response distributions 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 Discrete and continuous random variables Bernoulli Binomial Poisson Negative binomial Normal Chi-square and gamma Inverse Gaussian Overdispersion Exercises Exponential family responses and estimation 3.1 3.2 3.3 3.4 3.5 Exponential family The variance function Proof of the mean and variance expressions Standard distributions in the exponential family form Fitting probability functions to data Exercises v page ix 1 2 3 4 6 10 12 13 14 19 20 20 21 22 23 24 26 27 29 30 33 35 35 36 37 37 39 41

vi 4 5 6 7 Contents History and terminology of linear modeling Simple linear modeling Grouped and ungrouped data Transformations to normality and linearity Linear modeling 4.1 4.2 What does “linear” in linear model mean? 4.3 4.4 Multiple linear modeling The classical linear model 4.5 4.6 Least squares properties under the classical linear model 4.7 Weighted least squares 4.8 4.9 4.10 Categorical explanatory variables 4.11 Polynomial regression 4.12 Banding continuous explanatory variables Interaction 4.13 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 Generalized linear models 5.1 5.2 5.3 5.4 5.5 Maximum likelihood estimation 5.6 5.7 5.8 5.9 5.10 Further diagnostic tools 5.11 Model selection Conﬁdence intervals and prediction Assessing ﬁts and the deviance Testing the signiﬁcance of explanatory variables Residuals The generalized linear model Steps in generalized linear modeling Links and canonical links Offsets Exercises Models for count data 6.1 6.2 6.3 6.4 Poisson regression Poisson overdispersion and negative binomial regression Quasi-likelihood Counts and frequencies Exercises Categorical responses 7.1 Binary responses Logistic regression 7.2 42 42 43 43 44 46 47 47 48 49 51 53 54 55 55 56 58 60 61 62 64 64 65 66 66 67 70 71 74 77 79 80 80 81 81 89 94 96 96 97 97 98

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