Longevity-related Statistics and Their Perturbation Analysis: A Markov Chain, Matrix Calculus Approach

人类寿命一直是人口学研究的中心焦点。它反映了整个生命周期中死亡的强度、年龄模式和原因，从婴儿死亡到老年死亡以及介于两者之间的一切。这项研究项目的目的是开发和应用新的数学工具来对寿命进行敏感性分析。敏感性分析将变化的影响量化，这些新工具将使研究人员能够量化死亡率变化对衡量寿命的各种统计数据的影响。这些统计数据包括预期寿命、寿命变异指数以及特定死因对这些统计数据的贡献。敏感性分析在长寿研究中很重要，因为在过去的两个世纪里，由于医学、公共卫生和技术的前所未有的进步，人类的死亡率发生了巨大的变化。除了这些时间上的变化外，在任何时候，长寿模式在区域、国家和社会经济群体之间也不同。总之，这些长寿方面的变化对社会具有重要影响，它们决定了人口的年龄结构以及对医疗和社会服务的需求。这项研究开发的新分析方法将在长寿及其后果的研究中广泛使用。

Human longevity has always been a central focus of demographic research. It reflects the intensity, age pattern, and causes of mortality throughout the life cycle, from infant mortality to old-age mortality and everything in between. The objective of this research project is to develop and apply new mathematical tools for the sensitivity analysis of longevity. Sensitivity analysis quantifies the effects of changes, and these new tools will enable researchers to quantify the effects of changes in mortality on various statistics measuring longevity. Those statistics include life expectancy, indices of variation in longevity, and the contributions of particular causes of death to those statistics.Sensitivity analysis is important in the study of longevity because human mortality has changed dramatically over the last two centuries, due to unprecedented advances in medicine, public health, and technology. In addition to these temporal changes, at any one time, patterns of longevity also differ among regions, countries, and socio-economic groups. Together, these changes in longevity have important implications for society, determining, as they do, the age structure of populations and the needs for medical care and social services. The new analytical methods developed in this research will be widely useful in studies of changing longevity and its consequences.