Introduction 1

I Credibility models for claim frequencies 15

1 On the dependence induced by frequency credibility models 17

1 Introduction and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2 Poisson credibility models incorporating a priori classification . . . . . . . . 19

3 Statements S1-S3 in the model A1-A2 . . . . . . . . . . . . . . . . . . . . . 22

3.1 Stochastic order relations . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2 Statements S1-S2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.3 Positive dependence notions for random couples . . . . . . . . . . . 26

3.4 Statement S3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4 Dependence between annual claim numbers . . . . . . . . . . . . . . . . . . 29

4.1 Positive dependence notions for random vectors . . . . . . . . . . . . 29

4.2 Serial dependence for claim frequencies . . . . . . . . . . . . . . . . 31

5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2 On the stochastic increasingness of future claims in the B¨uhlmann linear

credibility premium 33

1 Introduction and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

1.1 Credibility theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

ii

1.2 GLM’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

1.3 Credibility theory and GLMM’s . . . . . . . . . . . . . . . . . . . . 35

1.4 Scope of the work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2 Preliminary results and concepts . . . . . . . . . . . . . . . . . . . . . . . . 37

2.1 Univariate stochastic dominance . . . . . . . . . . . . . . . . . . . . 37

2.2 Stochastic increasingness of Yit in the canonical parameter . . . . . . 39

2.3 Multivariate stochastic dominance . . . . . . . . . . . . . . . . . . . 39

2.4 Stochastic increasingness of Y i in the canonical parameter . . . . . 40

3 Exhaustive summary of past claims . . . . . . . . . . . . . . . . . . . . . . . 40

4 A posteriori distribution of the random effects . . . . . . . . . . . . . . . . . 41

5 Predictive distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

6 Linear credibility premium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3 Dependence in dynamic claim frequency credibility models 49

1 Introduction and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2 Poisson credibility models incorporating a priori risk classification . . . . . 51

3 Modelling heterogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.1 Model A3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.2 Model A4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.3 Model A5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.4 Model A6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.5 Model A7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4 Statement S1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.1 Stochastic order relations . . . . . . . . . . . . . . . . . . . . . . . . 55

4.2 Dependence concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.3 MTP2 functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.4 Proof of statement S1 . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.5 Statement S1 in the models A3-A7 . . . . . . . . . . . . . . . . . . . 60

5 Statement S2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

6 Statements S3 and S4 in models A5-A7 . . . . . . . . . . . . . . . . . . . . 65

7 Some particularities of the static model A3 . . . . . . . . . . . . . . . . . . 67

iii

8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4 Linear credibility models based on time series for claim counts 69

1 Introduction and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

2 Description of the data set . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3 Modelling through random effects . . . . . . . . . . . . . . . . . . . . . . . . 73

3.1 Description of the model . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.2 Multivariate Poisson-LogNormal distribution . . . . . . . . . . . . . 74

3.3 Application to Spanish panel data . . . . . . . . . . . . . . . . . . . 78

4 Comparison of linear credibility updating formulas . . . . . . . . . . . . . . 86

4.1 Derivation of linear credibility formulas . . . . . . . . . . . . . . . . 86

4.2 A posteriori correction according to age of claims . . . . . . . . . . 88

4.3 A posteriori correction according to a priori characteristics . . . . . 90

4.4 A posteriori correction according to the model used for series of claim

counts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

II Copula modelling 103

5 Bivariate archimedean copula modelling for loss-ALAE data in non-life

insurance 105

1 Introduction and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

1.1 Losses and their associated ALAE’s . . . . . . . . . . . . . . . . . . 105

1.2 Presentation of the ISO data set . . . . . . . . . . . . . . . . . . . . 106

1.3 Modelling loss-ALAE data with archimedean copulas . . . . . . . . . 107

1.4 Aim of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

1.5 Agenda . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

2 Archimedean copulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

2.1 Sklar’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

2.2 Archimedean family . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

3 Nonparametric estimation of the generator in presence of censored data . . 112

iv

3.1 General principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

3.2 Estimating Kendall’s tau . . . . . . . . . . . . . . . . . . . . . . . . 113

3.3 Genest-Rivest estimation procedure for the generator with complete

data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

3.4 Wang-Wells general estimation procedure for the generator in the

presence of censored data . . . . . . . . . . . . . . . . . . . . . . . . 114

3.5 Akritas estimation procedure for a bivariate distribution function under

censoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

4 Application to loss-ALAE modelling . . . . . . . . . . . . . . . . . . . . . . 117

4.1 Nonparametric estimation of the generator . . . . . . . . . . . . . . 117

4.2 Comparison with Dabroswka and Genest-Rivest estimations . . . . . 118

4.3 Graphical model selection procedure for the generator . . . . . . . . 120

4.4 Graphical representations . . . . . . . . . . . . . . . . . . . . . . . . 122

5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

Conclusion and future research 127

Bibliography 133

 

 

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