Computational and statistical methods for loss models

这项建议是为了与研究生和行业合作伙伴共同努力，建立保险损失的现代统计模型。ITL分为三个子项目。*贝叶斯可信度中的预测方法：经典可信度理论回答了以下两个问题：(1)一个风险类别中需要多少个观察才能完全基于其样本值来确定其溢价？(完全可信度)，以及(2)“如果不是完全可信度，如何在样本外信息中混合，以改进基于样本的风险类别保费估计器”(部分可信度)。在60‘S和70’S中，贝叶斯给出了这些问题的答案，重点是解析解、后验均值的线性估计和方差(可信区间)的渐近结果。*是时候用现代计算工具重新审视这个理论了。我们将GLMS用于在一般贝叶斯框架中设置的分段保单组合。除了GLM假设外，我们的先验分布和模型分布不需要是自然共轭的，也不需要对保费施加任何线性约束。后验分布和预测分布通过MCMC模拟进行评估，以近似积分，并用于回答上述两个可信度问题。*交互术语的机器学习技术：适用于保险投资组合的GLMS使用大量协变量(100)将保单划分为风险类别。这些协变量在被链接函数修改之前线性地进入GLM均值。引入非线性术语，例如协变量之间的相互作用，可能会给出更好的表示。选择最重要的相互作用成为一个非常高维度的问题。*在高维模型中使用正则化，以帮助自动选择变量。我们建议将保险GLMS推广为包含正则化的模型，如岭回归、套索、组套索或弹性网，并将它们与聚集了简单的基于树的模型的广义助推模型(GBM)进行比较。*保险GLM中的隐马尔可夫链：GLM是静态的，因为风险特征(协变量)是基于固定时间段的过去信息来确定投保人下一年的风险类别。例如，在汽车保险中，司机的风险分类可能取决于她/他最近3年的事故数量。这种分类只有在未来一年再次适用该模型时才会改变。*我们研究了一个随时间变化的损失模型，在该模型中，投保人的驾驶能力可能会发生变化，可能是由于最近发生事故后安全意识的增强，或者是在多年没有发生事故后不那么安全、过于自信的驾驶行为。我们提出了一个隐马尔可夫模型(HMM)，其中当前的行为(好的/坏的驾驶)对保险公司来说是不可观测的，但它对索赔数量/严重性的影响是。*

This proposal is for joint work with graduate students and industrial partners on modern statistical modeling for insurance losses. Itl divides in 3 subprojects.***Predictive methods in Bayesian credibility: Classical credibility theory answers the 2 following questions: (1) “how many observations are needed in a risk class before its premium can be based solely on its sample values?” (full credibility) , and (2) “if not it is not fully credible, how can out-of-sample information be mixed in to improve the risk class sample-based premium estimator” (partial credibility). In the 60's and 70's a Bayesian answer was given to these questions, with an emphasis on analytical solutions, linear estimators for the posterior mean and asymptotic results for the variance (confidence intervals).****It is time to revisit the theory using modern computational tools. We use GLMs for a segmented portfolio of insurance policies, set in a general Bayesian framework. Our prior and model distributions do not need to be natural conjugate, nor are any linearity constraints imposed on premiums, apart from the GLM assumption. The posterior and predictive distributions are evaluated through MCMC simulations, to approximate integrals, and used to answer the 2 credibility questions above plus much more. ***Machine learning techniques for interaction terms: The GLMs fitted to insurance portfolios use a large number of covariates (100+) to segregate policies into risk classes. These covariates enter the GLM-mean linearly, before being modified by a link function. The introduction of non-linear terms, such as interactions between covariates, may give a better representation. The choice of the most significant interactions becomes a very high dimensional problem. ***Regularization is used in high-dimensional models to help automatize variable selection. We propose to generalize insurance GLMs to include regularization, such as Ridge Regression, Lasso, Group-Lasso or Elastic Net and compare them to generalized boosted models (GBM), which aggregates simple tree-based models. ***Hidden Markov chains in insurance GLMs: GLMs are static, in the sense that the risk characteristics (covariates) are based on past information over a fixed period of time to determine a policyholder's risk classes during the next year. For instance, in auto insurance a driver's risk classification may depend on her/his number of accidents in the last 3 years. This classification can only change once the model is fitted again in a future year.****We study a time-dependent loss model where the policyholder's driving ability can change, perhaps due to an increased safety awareness following a recent accident, or a less safe, overconfident driving behaviour following years without accidents. We propose a hidden Markov model (HMM), where current behaviour (good/bad driving) is not observable for the insurer, but its impact on the number/severity of claims is.*********