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
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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
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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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