Computational and statistical methods for loss models

该提案旨在与研究生和工业合作伙伴共同研究保险损失的现代统计建模。贝叶斯可信度的预测方法：经典可信度理论回答了以下两个问题：（1）“在一个风险类别中需要多少个观测值才能完全基于其样本值来计算其保费？”(full可信度），以及（2）“如果不完全可信，如何将样本外信息混合进来，以改进基于样本的风险类别保费估计量”（部分可信度）。在60年代和70年代，贝叶斯回答了这些问题，重点是分析解决方案，后验均值的线性估计和方差的渐近结果（置信区间）。现在是时候使用现代计算工具重新审视理论了。我们使用GLM的一个分段的投资组合的保险政策，设置在一个一般的贝叶斯框架。我们的先验分布和模型分布不需要自然共轭，也没有任何线性约束强加于保费，除了GLM假设。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2个可信度问题以及更多问题。交互项的机器学习技术：适用于保险投资组合的GLM使用大量协变量（100+）将保单分为风险类别。这些协变量在被链接函数修改之前线性地进入GLM均值。引入非线性项，如协变量之间的相互作用，可以提供更好的表示。最重要的相互作用的选择成为一个非常高维的问题。在高维模型中使用正则化来帮助自动化变量选择。我们建议将保险GLM推广到包括正则化，例如岭回归，Lasso，Group-Lasso或弹性网络，并将它们与广义提升模型（GBM）进行比较，后者聚合了简单的基于树的模型。保险GLM中的隐马尔可夫链：GLM是静态的，在这个意义上，风险特征（协变量）是基于固定时间段内的过去信息，以确定保单持有人在下一年的风险类别。例如，在汽车保险中，驾驶员的风险分类可能取决于她/他在过去3年中的事故数量。我们研究了一个时间依赖性损失模型，其中投保人的驾驶能力可能会发生变化，这可能是由于最近发生事故后安全意识的提高，或者是由于多年没有发生事故后的不安全，过度自信的驾驶行为。我们提出了一个隐马尔可夫模型（HMM），目前的行为（好/坏驾驶）是不可观察的保险公司，但其对索赔的数量/严重程度的影响。

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.