The Makeham term is a crucial element in mortality modeling, representing a constant additive hazard that addresses background mortality factors unrelated to aging. Widely used in mortality analysis, this term enables the capture of risks not linked to age-related decline. This paper aims to explore the relationship between Makeham mortality models and competing risk frameworks, investigating how Makeham models can be analyzed within the context of competing risks. It provides insights into the mathematical properties, interpretation, and applicability of Makeham models in modeling mortality risks associated with various causes of death. Additionally, it demonstrates that competing risk models can be represented as mixture models, enhancing understanding of mortality dynamics. The contribution lies in showing that Makeham mortality models, when represented as mixtures, offer a straightforward specification that can accommodate unobserved heterogeneity and distinguish between senescent and extrinsic mortality. By expressing Makeham models as a convex combination of probability distributions, the paper allows the estimation of premature mortality profiles, particularly at older ages, where most deaths are assumed to be senescent. It also facilitates the estimation of senescent mortality, which is crucial for studying the aging process.

翻译：Makeham项是死亡率建模中的关键要素，它表示一种恒定的加性风险，用于处理与衰老无关的背景死亡率因素。该术语广泛应用于死亡率分析，能够捕捉与年龄相关衰退无关的风险。本文旨在探究Makeham死亡率模型与竞争风险框架之间的关系，研究如何在竞争风险背景下分析Makeham模型，并深入阐释其在建模多种死因死亡率风险时的数学性质、解释方式及适用性。此外，本文证明竞争风险模型可表示为混合模型，从而增强对死亡率动态变化的理解。主要贡献在于：当Makeham死亡率模型以混合形式表示时，能够提供一种简洁的设定，既可容纳未观测到的异质性，又能区分衰老性死亡与外源性死亡。通过将Makeham模型表示为概率分布的凸组合，本文使得对过早死亡风险的分析成为可能，特别是在高龄阶段——此时大多数死亡被假定为衰老相关。同时，该研究还有助于估计衰老性死亡率，这对研究衰老过程至关重要。