We study tail estimation in Pareto-like settings for datasets with a high percentage of randomly right-censored data, and where some expert information on the tail index is available for the censored observations. This setting arises for instance naturally for liability insurance claims, where actuarial experts build reserves based on the specificity of each open claim, which can be used to improve the estimation based on the already available data points from closed claims. Through an entropy-perturbed likelihood, we derive an explicit estimator and establish a close analogy with Bayesian methods. Embedded in an extreme value approach, asymptotic normality of the estimator is shown, and when the expert is clair-voyant, a simple combination formula can be deduced, bridging the classical statistical approach with the expert information. Following the aforementioned combination formula, a combination of quantile estimators can be naturally defined. In a simulation study, the estimator is shown to often outperform the Hill estimator for censored observations and recent Bayesian solutions, some of which require more information than usually available. Finally we perform a case study on a motor third-party liability insurance claim dataset, where Hill-type and quantile plots incorporate ultimate values into the estimation procedure in an intuitive manner.

中文翻译：

---

使用删失数据和专家信息的组合尾部估计

我们研究 Pareto-like 设置中的尾部估计，用于随机右删失数据百分比较高的数据集，其中一些关于尾索引的专家信息可用于删失观察。例如，对于责任保险索赔，这种设置自然会出现，其中精算专家根据每个未结索赔的特殊性建立准备金，这可用于根据已结索赔的现有数据点改进估计。通过熵扰动的似然，我们推导出显式估计量并建立与贝叶斯方法的类比。嵌入极值方法，显示了估计量的渐近正态性，当专家有洞察力时，可以推导出一个简单的组合公式，将经典统计方法与专家信息联系起来。遵循上述组合公式，自然可以定义分位数估计量的组合。在模拟研究中，对于删失观测和最近的贝叶斯解决方案，估计量通常优于 Hill 估计量，其中一些需要比通常可用的更多信息。最后，我们对汽车第三方责任保险索赔数据集进行案例研究，其中希尔型和分位数图以直观的方式将最终值纳入估计程序。