Background: The Makeham term serves as a fundamental component in mortality modeling, offering a constant additive hazard that accounts for background mortality factors usually unrelated to the aging process. This term, widely employed in mortality analysis, provides a crucial mechanism for capturing mortality risks unrelated to age-related deterioration. Objective: The objective of this paper is to explore how Makeham models, which are widely used for studying mortality, can be understood and analyzed within the context of competing risks. The paper seeks to provide insights into the mathematical properties, interpretation, and applicability of Makeham models in modeling age-dependent and age-independent mortality risks. Additionally, the paper aims to demonstrate formally that competing-risk models can be represented as mixture models, thereby facilitating a deeper understanding of risk-specific mortality dynamics. Contribution: Expressing competing-risk models as mixtures aids identifying the overall and agespecific share of deaths according to each of the competing risks. In particular, Makeham mortality models, when represented as mixtures, provide, first, a semantically and computationally convenient platform to disentangle age-dependent from age-independent mortality, and second, a straightforward specification that can easily be extended to account for unobserved heterogeneity. By expressing Makeham models as a convex combination of probability distributions, we are able to estimate the age-profile of age-independent mortality, especially at the oldest ages, at which we intuitively assume that most deaths are age-dependent (senescent). We are also able to estimate the senescent mortality component, which is the one to focus on when studying the aging process and its characteristics.

中文翻译：

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Makeham 死亡率模型作为混合物：通过竞争风险框架推进死亡率估计（作者：Silvio Cabral Patricio、Trifon Missov）


背景：Makeham 项是死亡率模型的基本组成部分，提供了恒定的附加风险，可以解释通常与衰老过程无关的背景死亡率因素。该术语广泛用于死亡率分析，为捕获与年龄相关恶化无关的死亡风险提供了关键机制。目的：本文的目的是探讨如何在竞争风险的背景下理解和分析广泛用于研究死亡率的 Makeham 模型。本文旨在深入了解 Makeham 模型在建模年龄相关和年龄无关的死亡风险方面的数学特性、解释和适用性。此外，本文旨在正式证明竞争风险模型可以表示为混合模型，从而促进对特定风险死亡率动态的更深入理解。贡献：将竞争风险模型表示为混合物有助于根据每种竞争风险确定总体和特定年龄的死亡比例。特别是，当以混合物表示时，Makeham 死亡率模型首先提供了一个语义上和计算上方便的平台，用于区分年龄依赖性死亡率和年龄无关死亡率，其次提供了一个简单的规范，可以轻松扩展以解释未观察到的异质性。通过将 Makeham 模型表示为概率分布的凸组合，我们能够估计与年龄无关的死亡率的年龄分布，特别是在最大年龄，我们直观地假设大多数死亡与年龄相关（衰老）。 我们还能够估计衰老死亡率部分，这是研究衰老过程及其特征时需要重点关注的部分。