Mack's distribution-free chain ladder reserving model belongs to the most popular approaches in non-life insurance mathematics. Proposed to determine the first two moments of the reserve, it does not allow to identify the whole distribution of the reserve. For this purpose, Mack's model is usually equipped with a tailor-made bootstrap procedure. Although widely used in practice to estimate the reserve risk, no theoretical bootstrap consistency results exist that justify this approach. To fill this gap in the literature, we adopt the framework proposed by Steinmetz and Jentsch (2022) to derive asymptotic theory in Mack's model. By splitting the reserve into two parts corresponding to process and estimation uncertainty, this enables - for the first time - a rigorous investigation also of the validity of the Mack bootstrap. We prove that the (conditional) distribution of the asymptotically dominating process uncertainty part is correctly mimicked by Mack's bootstrap if the parametric family of distributions of the individual development factors is correctly specified. Otherwise, this is not the case. In contrast, the (conditional) distribution of the estimation uncertainty part is generally not correctly captured by Mack's bootstrap. To tackle this, we propose an alternative Mack-type bootstrap, which is designed to capture also the distribution of the estimation uncertainty part. We illustrate our findings by simulations and show that the newly proposed alternative Mack bootstrap performs superior to the Mack bootstrap.

翻译：Mack的无分布链梯级准备金模型属于非寿险数学中最常用的方法之一。尽管该方法最初用于确定准备金的前两阶矩，却无法识别准备金的完整分布。为此，Mack模型通常配备定制的自助法（bootstrap）。尽管此法在实践中被广泛用于估计准备金风险，但至今尚无理论上的自助法相合性结果来验证其有效性。为填补这一文献空白，我们采用Steinmetz与Jentsch（2022）提出的框架推导Mack模型中的渐近理论。通过将准备金分解为过程不确定性与估计不确定性两部分，本研究首次实现对Mack自助法有效性的严格考察。我们证明，若个体进展因子的参数分布族被正确设定，Mack自助法能正确模拟渐近主导的过程不确定性部分的（条件）分布；反之则不然。相比之下，Mack自助法通常无法正确捕捉估计不确定性部分的（条件）分布。为解决这一问题，我们提出一种改进的Mack型自助法，该方法旨在同时捕获估计不确定性部分的分布。通过模拟实验验证发现，新提出的改进型Mack自助法表现优于原始Mack自助法。