The design of an optimal Bonus-Malus System based on the Sichel distribution

This chapter presents the design of an optimal Bonus-Malus System (BMS) using the Sichel distribution to model the claim frequency distribution. This system is proposed as an alternative to the optimal BMS obtained by the traditional Negative Binomial model [19]. The Sichel distribution has a thicker tail than the Negative Binomial distribution and it is considered as a plausible model for highly dispersed count data. We also consider the optimal BMS provided by the Poisson-Inverse Gaussian distribution (PIG), which is a special case of the Sichel distribution. Furthermore, we develop a generalised BMS that takes into account both the a priori and a posteriori characteristics of each policyholder. For this purpose we consider the generalised additive models for location, scale and shape (GAMLSS) in order to use all available information in the estimation of the claim frequency distribution. Within the framework of the GAMLSS we propose the Sichel GAMLSS for assessing claim frequency as an alternative to the Negative Binomial Type I (NBI) regression model used by Dionne and Vanasse [9, 10]. We also consider the NBI and PIG GAMLSS for assessing claim frequency

Statistical procedure for improving the precision of the measurement of the essential work of fracture of thin sheets

The precision of the measurement of the essential work of fracture requires an accurate determination of the critical ligament length below which the ligament is in a mixed mode stress state and the failure mechanisms considerably change. A statistical procedure is proposed which allows to determine easily and accurately this critical ligament length. After rejection of the specimens having too small ligament lengths, w(e) is obtained by extrapolation of the remaining data for zero ligament length. The number of data points from which w, is calculated is also shown to strongly influence the precision of the measurement. The procedure is applied to the measurement of the toughness of sheets of Al and Zn alloys and of a low density polyethylene

Initial premium, aggregate claims and distortion risk measures in XL reinsurance with reinstatements

With reference to risk adjusted premium principle, in this paper we study excess of loss reinsurance with reinstatements in the case in which the aggregate claims are generated by a discrete distribution. In particular, we focus our study on conditions ensuring feasibility of the initial premium, for example with reference to the limit on the payment of each claim. Comonotonic exchangeability shows the way forward to a more general definition of the initial premium: some properties characterizing the proposed premium are presented