Abstract
Abstract 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.
Highlights
Numerous authors have studied attributes of mortality data, for instance, Cairns et al (2008) proposed several stylised facts of mortality data
To measure the strength of long memory, we estimate H using all three estimation methods presented (R/S, detrended fluctuation analysis (DFA) and periodogram regression (PR)) and we only report the results of the range analysis (R/S) in section 2.3.1 since the other two methods provide very similar results, and this is evidenced in the hosted Supplementary Material on the AAS website
It is analogous to stating the female mortality in these age groups has much stronger second order autocorrelation relationships over the decades compared to males globally. Such a stylised feature can be incorporated to improve mortality projection, it is clear that based on these findings models of mortality for males and females and in different age bands should be enhanced to incorporate these differences in long memory persistence patterns in order to better improve the mortality forecasting and life table construction
Summary
The main contribution of this work is to present evidence for the presence of persistence or long memory in mortality data and to study the behaviour of such features in mortality rates from 16 countries cross-classified by gender and age groups. We conclude the paper by demonstrating how the functional feature extraction of multifractal Hurst exponent curves can be used to perform clustering of mortality experience by age, country and gender in an advanced non-linear discriminant method based on kernel k-means, a modern machine learning variant of traditional k-means, in which function space clustering is applied We demonstrate that such decompositions allow us to group countries and age groups into three classes of mortality structures as characterised by their long memory characteristics.
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