Panel Count Data with Dynamic Random Effects in Actuarial Sciences

我的研究项目是关于一般保险（火灾，事故和伤亡）的数学发展。其主要思想是建立一个保险公司的索赔数量模型，其中同一个被保险人的所有合同之间的依赖关系，称为面板数据结构，将在建模中提出。 为此，研究计划分为两个独立的轴： 1.在初步应用于真实的数据面板之前，需要对时间序列进行计数的新统计发展的理论方法。将测试各种形式的条件分布（如障碍或零膨胀模型），并验证各种形式的潜在动态过程（具有大量分形过程或使用其他形式过程的多重分形模型的推广）。将使用和推广新的估计技术和新的时间序列模型，例如粒子滤波器近似或线性卡尔曼滤波器等滤波技术。由于其快速性，多重分形模型也可以用于一些重要抽样方法。 2.一种基于奖惩系统（BMS）的实用方法，其中，我们假设BMS扮演这个角色，而不是使用高级统计模型，该模型在同一个被保险人的所有合同之间创建时间依赖性。在这种创新的方法中，一个新的标准，提出了基于竞争的角度。 事实上，保险公司必须提供足够高的保费来覆盖风险，但低于竞争对手。 我们将选择最好的BMS在这个标准，而不是使用经典的信息标准的基础上对数似然统计。

My research program is about mathematical developments for general insurance (Fire, Accident & Casualty). The main idea is to model the number of claims to an insurer where the dependence between all the contracts of the same insured, called panel data structure, will be put forward in the modeling. To do this, the research program is divided into two independent axes: 1. A theoretical approach that requires new statistical developments on time series for count, before the preliminary application to real data panel. Various forms of conditional distributions (such as hurdle or zero-inflated models) will be tested, and various forms of the underlying dynamic process will also be verified (generalization of the multifractal models with a larger number of fractal processes, or using other forms of processes). New estimation techniques and new time series models will be used and generalized, such as filtering techniques, like particle filter approximation or a linear Kalman filter, for example. Because of their high speed, multifractal models can also be used in some importance sampling methods. 2. A practical approach based on bonus-malus systems (BMS) where, instead of working with an advanced statistical model that creates time dependence between all the contracts of the same insured, we assume that a BMS plays this role. In this innovative approach, a new criterion is proposed based on a competitive angle. Indeed, the insurer must offer a premium high enough to cover the risk, but lower than that of the competitors. We will select the best BMS on this criterion instead of using classic information criterion based on log-likelihood statistics.