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、意外和意外伤害)的数学发展。其主要思想是对保险公司的索赔数量进行建模，其中同一被保险人的所有合同之间的相关性将在建模中提出，称为面板数据结构。为此，研究计划被分成两个独立的轴：1.理论方法，要求在初步应用于实际数据面板之前，对时间序列进行统计的新的统计发展。将测试各种形式的条件分布(如栅栏或零膨胀模型)，还将验证各种形式的基本动态过程(推广具有更多数量的分形过程的多重分形模型，或使用其他形式的过程)。将使用和推广新的估计技术和新的时间序列模型，例如像粒子滤波近似或线性卡尔曼滤波之类的滤波技术。由于多重分形模型具有较高的运算速度，因此也可以用于一些重要的采样方法中。一种基于奖金-奖金系统(BMS)的实用方法，其中，我们假设BMS扮演这一角色，而不是使用高级统计模型来创建相同投保人的所有合同之间的时间相关性。在这一创新的方法中，基于竞争的角度提出了一个新的标准。事实上，保险公司必须提供足够高的保费来覆盖风险，但低于竞争对手的保费。我们将在此准则下选择最优的BMS，而不是使用基于对数似然统计的经典信息准则。

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.