# Modelling dependence in actuarial science, with emphasis on credibility theory and copulas

Abstract (Summary)

One basic problem in statistical sciences is to understand the relationships among multivariate outcomes. Although it remains an important tool and is widely applicable,
the regression analysis is limited by the basic setup that requires to identify one dimension of the outcomes as the primary measure of interest (the "dependent"
variable) and other dimensions as supporting this variable (the "explanatory" variables). There are situations where this relationship is not of primary interest.
For example, in actuarial sciences, one might be interested to see the dependence between annual claim numbers of a policyholder and its impact on the premium
or the dependence between the claim amounts and the expenses related to them. In such cases the normality hypothesis fails, thus Pearson's correlation or concepts based
on linearity are no longer the best ones to be used. Therefore, in order to quantify the dependence between non-normal outcomes one needs different statistical tools,
such as, for example, the dependence concepts and the copulas.
This thesis is devoted to modelling dependence with applications in actuarial sciences and is divided in two parts: the first one concerns dependence in frequency
credibility models and the second one dependence between continuous outcomes. In each part of the thesis we resort to different tools, the stochastic orderings
(which arise from the dependence concepts), and copulas, respectively.
During the last decade of the 20th century, the world of insurance was confronted with important developments of the a posteriori tarification, especially in the
field of credibility. This was dued to the easing of insurance markets in the European Union, which gave rise to an advanced segmentation. The first important
contribution is due to Dionne & Vanasse (1989), who proposed a credibility model which integrates a priori and a posteriori information on an individual basis.
These authors introduced a regression component in the Poisson counting model in order to use all available information in the estimation of accident frequency.
The unexplained heterogeneity was then modeled by the introduction of a latent variable representing the influence of hidden policy characteristics. The vast majority
of the papers appeared in the actuarial literature considered time-independent (or static) heterogeneous models. Noticeable exceptions include the pioneering papers
by Gerber & Jones (1975), Sundt (1988) and Pinquet, GuillĂ©n & BolancĂ© (2001, 2003). The allowance for an unknown underlying random parameter
that develops over time is justified since unobservable factors influencing the driving abilities are not constant. One might consider either shocks (induced by
events like divorces or nervous breakdown, for instance) or continuous modifications (e.g. due to learning effect).
In the first part we study the recently introduced models in the frequency credibility theory, which can be seen as models of time series
for count data, adapted to actuarial problems. More precisely we will examine the kind of dependence induced among annual claim numbers by the introduction of random
effects taking unexplained heterogeneity, when these random effects are static and time-dependent. We will also make precise the effect of reporting claims on the
a posteriori distribution of the random effect. This will be done by establishing some stochastic monotonicity property of the a posteriori distribution
with respect to the claims history. We end this part by considering different models for the random effects and computing the a posteriori corrections of the
premiums on basis of a real data set from a Spanish insurance company.
Whereas dependence concepts are very useful to describe the relationship between multivariate outcomes, in practice (think for instance to the computation of reinsurance
premiums) one need some statistical tool easy to implement, which incorporates the structure of the data. Such tool is the copula, which allows the construction of multivariate
distributions for given marginals. Because copulas characterize the dependence structure of random vectors once the effect of the marginals has been factored out,
identifying and fitting a copula to data is not an easy task. In practice, it is often preferable to restrict the search of an appropriate copula to some reasonable
family, like the archimedean one. Then, it is extremely useful to have simple graphical procedures to select the best fitting model among some competing alternatives
for the data at hand.
In the second part of the thesis we propose a new nonparametric estimator for the generator, that takes into account the particularity of the data, namely censoring and truncation.
This nonparametric estimation then serves as a benchmark to select an appropriate parametric archimedean copula. This selection procedure will be illustrated
on a real data set.
Bibliographical Information:

Advisor:

School:Université catholique de Louvain

School Location:Belgium

Source Type:Master's Thesis

Keywords:archimedean copula linear credibility predictor quadratic loss positive dependence stochastic orderings autoregressive time series of count data generalized mixed models theory bivariate censored nonparametric estimation

ISBN:

Date of Publication:08/19/2005