Modal Interval Probability: Application to Bonus-Malus Systems

Abstract

Classical intervals have been a very useful tool to analyze uncertain and imprecise models, in spite of operative and interpretative shortcomings. The recent introduction of modal intervals helps to overcome those limitations. In this paper, we apply modal intervals to the field of probability, including properties and axioms that form a theoretical framework applied to the Markovian analysis of Bonus-Malus systems in car insurance. We assume that the number of claims is a Poisson distribution and in order to include uncertainty in the model, the claim frequency is defined as a modal interval; therefore, the transition probabilities are modal interval probabilities. Finally, the model is exemplified through application to two different types of Bonus-Malus systems, and the attainment of uncertain long-run premiums expressed as modal intervals.

Fuzzy Markovian Bonus-Malus Systems in Non-Life Insurance

“…However, in a more realistic approach, some authors like [15] use intervals to quantify the uncertainty about the parameter of a distribution function that governs a risk variable. This is also the case within a BMS framework of [11], who model the uncertainty about by means of a modal interval. One extended way to combine randomness and uncertainty of parameters of distribution functions consists in modeling these parameters as FNs.…”

“…Therefore, these BMSs are Markovian. For that reason, the academic literature on BMSs uses extensively MCs for their modeling ( [1,[5][6][7][8][9][10][11]). Therefore, a key question in a BMS is fitting the value of the one-step transition probability matrix.…”

“…Since any interval can be seen as the -cut of a FN, even in the case of improper intervals ( [29]), in this work we consider that is fitted by means of a FN and, more particularly, by a triangular fuzzy number (TFN). So, this paper builds up a framework to model Markovian BMSs that embed the standard case, where the risk parameter is crisp, but also the method developed in [11] that quantifies this parameter as a modal interval. Standard BMSs provide point values for the stationary state and the mean asymptotic premium.…”

“…Standard BMSs provide point values for the stationary state and the mean asymptotic premium. Modal BMSs as introduced in [11] allow obtaining these variables as modal intervals whose lower and upper bounds may be understood as pessimistic/optimistic scenarios. Our method generalizes both types of BMSs since it quantifies variables related to BMS as FNs.…”

“…BMSs consider the number of claims, , as a discrete RV. In our paper, as it is commonplace in actuarial literature, is supposed to follow a Poisson distribution with parameter , ~Po(l), ( [1,[7][8][9][10][11][12]). Therefore:…”