Abstract
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.
Highlights
The parametric accelerated failure time (AFT) model has been well established in medical statistics and other applications (Kalbfleisch and Prentice 2002) for decades
In Model (1), the operational time function φ links e to the observed settlement delay depending on the accident date
We introduced a new hazard model that allows for operational time in right-truncated data as present in run-off triangles
Summary
The parametric accelerated failure time (AFT) model has been well established in medical statistics and other applications (Kalbfleisch and Prentice 2002) for decades. Based on the idea of chain-ladder, different multiplicative models with independent effects of accident date and settlement delay were introduced in Kremer (1982); Kuang et al (2009); Renshaw and Verrall (1998), and Verrall (1991) Aside from these publications, the greater part of the research on claims reserving can be summarized into two streams: a Poisson process approach and a two-dimensional kernel estimation approach for truncated data. In a broader statistical context, the problem was introduced as “in-sample forecasting” (Mammen et al 2015) and said papers applied their results to forecasting problems beyond actuarial research These articles have in common that no distributional assumptions are made and that kernel estimation is performed under the assumption of a structural model for the joint density or conditional hazard rate. Since this particular triangular data structure can be expressed as truncated data, a counting process survival model lends itself to our cause
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