Smooth backfitting was first introduced in an additive regression setting via a direct projection alternative to the classic backfitting method by Buja, Hastie and Tibshirani. This paper translates the original smooth backfitting concept to a survival model considering an additively structured hazard. The model allows for censoring and truncation patterns occurring in many applications such as medical studies or actuarial reserving. Our estimators are shown to be a projection of the data into the space of multivariate hazard functions with smooth additive components. Hence, our hazard estimator is the closest nonparametric additive fit even if the actual hazard rate is not additive. This is different to other additive structure estimators where it is not clear what is being estimated if the model is not true. We provide full asymptotic theory for our estimators. We provide an implementation the proposed estimators that show good performance in practice even for high dimensional covariates.

翻译：平滑回拟最初由Buja、Hastie和Tibshirani在加性回归框架中提出，作为经典回拟方法的直接投影替代方案。本文将原始平滑回拟概念迁移至考虑加性结构风险的生存模型。该模型允许处理医学研究或精算准备金等众多应用中出现的删失与截断模式。我们证明所提出的估计量是对数据在具有光滑加性成分的多变量风险函数空间上的投影。因此，即使实际风险率非加性，我们的风险估计量仍是最优的非参数加性拟合。这与其它加性结构估计量不同——在模型不成立时，后者无法明确估计目标。我们建立了估计量的完整渐近理论，并提供了实现方法。数值实验表明，即使面对高维协变量，所提估计量仍具有良好实践性能。