Copula-based Dependence In Car Insurance With Excess Zeros

In automobile insurance, a bonus-malus system calculates the premium applicable to particular contract as a base premium, and is adjusted by a quantity (the bonus or malus) which depends on previous claim experience. In general, the claims of the policyholders between contracts are assumed to be independent, but correlation of the claims of the same individual is permitted.A typical dependency is serial correlation between previous and current claim numbers. This paper develops models aimed at more accurate estimation and prediction of the number of claims in insurance based on copula models, in which the dependency between the numbers of the claims of two successive periods is assumed to be directly or indirectly modeled by discrete or continuous marginal distributions, seperately. The first model is to fit the two successive claims by a bivariate copula with discrete marginal distribution. The dependency between the numbers of the claims of two successive periods can be directly displayed by the estimator of the copula parameter. The second copula is used to model the random effects of the conjoint numbers of the successive claims with continuous marginal distribution. Random effects, which are involved into the mean value of the number of claims and whose dependency is able to indirectly reflect the dependency between the number of the claims of two successive periods, partly exhibit the proneness of the claims happened. The ordinary maximum likelihood is applied to estimate the parameters of the discrete copula model. And the two-step procedure is used to estimate the parameters in the second case, that is, the first is to estimate the marginals and the copula parameter, and then the nonparametric estimation is given for the unobservable random effect variables.In most of literature, the number of the claims is often Poisson distributed and assumed to be independent of the class occupied by the policyholder. In the third copula of this paper, an alternative method is proposed to model the dependency of the class occupied during any period and the claim counts of the same period by using copula function, which illustrates the differenct risk proneness of policyholders in different classes and provides the theoretical foundation for the fact that policyholders in different classes are charged different premiums by insurers. Ordinary maximum likelihood is employed to obtain the estimations of model parameters and copula parameters."Zero-inflated phenomenon" is taken into account in the above three copula models. For every model proposed in this paper, the asymptotic properties of the estimators are established as well, furthermore, simulations are performed to illustrate and assess the proposed models and methodologies.The first highlight of this paper is that the bivariate copula model with zero-inflated is used to model the dependency between the numbers of the claims of two successive periods. Secondly, the copula function with continuous marginal is used model the random effects of the number of claims in successive periods. Thirdly, the dependency of the class occupied during any period and the claim counts of the same period is discussed and modeled. Finally, the zero-inflated model is taken into account in the bivariable copula functions. The new model and estimation method proposed in our paper will provide a potential contribution to the more accurate estimation and prediction of the number of the claims in actuarial science.