In this paper, a new class of composite model is proposed for modeling actuarial claims data of mixed sizes. The model is developed using the Stoppa distribution and a mode-matching procedure. The use of the Stoppa distribution allows for more flexibility over the thickness of the tail, and the mode-matching procedure gives a simple derivation of the model compositing with a variety of distributions. In particular, the Weibull–Stoppa and the Lognormal–Stoppa distributions are investigated. Their performance is compared with existing composite models in the context of the well-known Danish fire insurance data-set. The results suggest the composite Weibull–Stoppa model outperforms the existing composite models in all seven goodness-of-fit measures considered.

In this paper, a new methodology based on the use of the inverse of the circular tangent function that allows us to add a scale parameter (say α) to an initial survival function is presented. The latter survival function is determined as limiting case when α tends to zero. By choosing as parent the classical Pareto survival function, the Pareto ArcTan (PAT) distribution is obtained. After providing a comprehensive analysis of its statistical properties, theoretical results with reference to insurance are illustrated. Its performance is compared, by means of the well-known Norwegian fire insurance data, with other existing heavy-tailed distributions in the literature such as Pareto, Stoppa, Shifted Lognormal, Inverse Gamma and Fréchet distributions.

In the classical bonus-malus system the premium assigned to each policyholder is based only on the number of claims made without having into account the claims size. Thus, a policyholder who has declared a claim that results in a relatively small loss is penalised to the same extent as one who has declared a more expensive claim. Of course, this is seen unfair by many policyholders. In this paper, we study the factors that affect the number of claims in car insurance by using a trivariate discrete distribution. This approach allows us to discern between three types of claims depending wether the claims are above, between or below certain thresholds. Therefore, this model implements the two fundamental random variables in this scenario, the number of claims as well as the amount associated with them. In addition, we introduce a trivariate prior distribution conjugated with this discrete distribution that produce credibility bonus-malus premiums that satisfy appropriate traditional transition rules. A practical example based on real data is shown to examine the differences with respect to the premiums obtained under the traditional system of tarification.

In this paper, the inverse gamma power series (IGPS) class of distributions asymmetric is introduced. This family is obtained by compounding inverse gamma and power series distributions. We present the density, survival and hazard functions, moments and the order statistics of the IGPS. Estimation is first discussed by means of the quantile method. Then, an EM algorithm is implemented to compute the maximum likelihood estimates of the parameters. Moreover, a simulation study is carried out to examine the effectiveness of these estimates. Finally, the performance of the new class is analyzed by means of two asymmetric real data sets.

A comprehensive auto insurance policy usually provides the broadest protection for the most common events for which the policyholder would file a claim. On the other hand, some insurers offer extended third-party car insurance to adapt to the personal needs of every policyholder. The extra coverage includes cover against fire, natural hazards, theft, windscreen repair, and legal expenses, among some other coverages that apply to specific events that may cause damage to the insured’s vehicle. In this paper, a multivariate distribution, based on a conditional specification, is proposed to account for different numbers of claims for different coverages. Then, the premium is computed for each type of coverage separately rather than for the total claims number. Closed-form expressions are given for moments and cross-moments, parameter estimates, and for a priori premiums when different premiums principles are considered. In addition, the severity of claims can be incorporated into this multivariate model to derive multivariate claims’ severity distributions. The model is extended by developing a zero-inflated version. Regression models for both multivariate families are derived. These models are used to fit a real auto insurance portfolio that includes five types of coverage. Our findings show that some specific covariates are statistically significant in some coverages, yet they are not so for others.