We introduce a continuous-time framework for the prediction of outstanding liabilities, in which chain-ladder development factors arise as a histogram estimator of a cost-weighted hazard function running in reversed development time. We use this formulation to show that under our assumptions on the individual data chain-ladder is consistent. Consistency is understood in the sense that both the number of observed claims grows to infinity and the level of aggregation tends to zero. We propose alternatives to chain-ladder development factors by replacing the histogram estimator with kernel smoothers and by estimating a cost-weighted density instead of a cost-weighted hazard. Finally, we provide a real-data example and a simulation study confirming the strengths of the proposed alternatives.

We introduce a generalization of the one-dimensional accelerated failure time model allowing the covariate effect to be any positive function of the covariate. This function and the baseline hazard rate are estimated nonparametrically via an iterative algorithm. In an application in non-life reserving, the survival time models the settlement delay of a claim and the covariate effect is often called operational time. The accident date of a claim serves as covariate. The estimated hazard rate is a nonparametric continuous-time alternative to chain-ladder development factors in reserving and is used to forecast outstanding liabilities. Hence, we provide an extension of the chain-ladder framework for claim numbers without the assumption of independence between settlement delay and accident date. Our proposed algorithm is an unsupervised learning approach to reserving that detects operational time in the data and adjusts for it in the estimation process. Advantages of the new estimation method are illustrated in a data set consisting of paid claims from a motor insurance business line on which we forecast the number of outstanding claims.

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 from 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 propose an implementation of estimators that shows good performance in practice.

We introduce a continuous-time framework for the prediction of outstanding liabilities, in which chain-ladder development factors arise as a histogram estimator of a cost-weighted hazard function running in reversed development time. We use this formulation to show that under our assumptions on the individual data chain-ladder is consistent. Consistency is understood in the sense that both the number of observed claims grows to infinity and the level of aggregation tends to zero. We propose alternatives to chain-ladder development factors by replacing the histogram estimator with kernel smoothers and by estimating a cost-weighted density instead of a cost-weighted hazard. Finally, we provide a real-data example and a simulation study confirming the strengths of the proposed alternatives.