Understanding core statistical properties and data features in mortality data are fundamental to the development of machine learning methods for demographic and actuarial applications of mortality projection. The study of statistical features in such data forms the basis for classification, regression and forecasting tasks. In particular, the understanding of key statistical structure in such data can aid in improving accuracy in undertaking mortality projection and forecasting when constructing life tables. The ability to accurately forecast mortality is a critical aspect for the study of demography, life insurance product design and pricing, pension planning and insurance-based decision risk management. Though many stylised facts of mortality data have been discussed in the literature, we provide evidence for a novel statistical feature that is pervasive in mortality data at a national level that is as yet unexplored. In this regard, we demonstrate in this work a strong evidence for the existence of long memory features in mortality data, and second that such long memory structures display multifractality as a statistical feature that can act as a discriminator of mortality dynamics by age, gender and country. To achieve this, we first outline the way in which we choose to represent the persistence of long memory from an estimator perspective. We make a natural link between a class of long memory features and an attribute of stochastic processes based on fractional Brownian motion. This allows us to use well established estimators for the Hurst exponent to then robustly and accurately study the long memory features of mortality data. We then introduce to mortality analysis the notion from data science known as multifractality. This allows us to study the long memory persistence features of mortality data on different timescales. We demonstrate its accuracy for sample sizes commensurate with national-level age term structure historical mortality records. A series of synthetic studies as well a comprehensive analysis of real mortality death count data are studied in order to demonstrate the pervasiveness of long memory structures in mortality data, both mono-fractal and multifractal functional features are verified to be present as stylised facts of national-level mortality data for most countries and most age groups by gender. We conclude by demonstrating how such features can be used in kernel clustering and mortality model forecasting to improve these actuarial applications.

中文翻译：

---

死亡率数据中持久性和长记忆的统计特征

了解死亡率数据中的核心统计特性和数据特征对于开发用于死亡率预测的人口统计和精算应用的机器学习方法至关重要。对此类数据中统计特征的研究构成了分类、回归和预测任务的基础。特别是，了解此类数据中的关键统计结构有助于提高在构建生命表时进行死亡率预测和预测的准确性。准确预测死亡率的能力是研究人口学、人寿保险产品设计和定价、养老金规划和基于保险的决策风险管理的关键方面。尽管文献中已经讨论了许多关于死亡率数据的程式化事实，我们提供了一种新的统计特征的证据，该特征在国家一级的死亡率数据中普遍存在，但尚未探索。在这方面，我们在这项工作中证明了死亡率数据中存在长记忆特征的有力证据，其次，这种长记忆结构显示出多重分形性作为一种统计特征，可以作为年龄、性别和死亡率动态的判别器。国家。为了实现这一点，我们首先从估计器的角度概述了我们选择表示长记忆持久性的方式。我们在一类长记忆特征和基于分数布朗运动的随机过程属性之间建立了自然联系。这使我们能够使用完善的赫斯特指数估计量，然后稳健而准确地研究死亡率数据的长记忆特征。然后，我们将数据科学中称为多重分形的概念引入死亡率分析。这使我们能够研究不同时间尺度上死亡率数据的长记忆持久性特征。我们证明了其与国家级年龄期限结构历史死亡率记录相称的样本量的准确性。为了证明长记忆结构在死亡率数据中的普遍性，研究了一系列综合研究以及对真实死亡率死亡计数数据的综合分析，单分形和多分形功能特征均被证实为大多数国家和大多数按性别分列的年龄组的国家级死亡率数据的程式化事实。最后，我们展示了如何在内核聚类和死亡率模型预测中使用这些特征来改进这些精算应用程序。