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Extension, Compression, and Beyond: A Unique Classification System for Mortality Evolution Patterns

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Demography

Abstract

A variety of literature addresses the question of how the age distribution of deaths changes over time as life expectancy increases. However, corresponding terms such as extension, compression, or rectangularization are sometimes defined only vaguely, and statistics used to detect certain scenarios can be misleading. The matter is further complicated because mixed scenarios can prevail, and the considered age range can have an impact on observed mortality patterns. In this article, we establish a unique classification framework for realized mortality scenarios that allows for the detection of both pure and mixed scenarios. Our framework determines whether changes of the deaths curve over time show elements of extension or contraction; compression or decompression; left- or right-shifting mortality; and concentration or diffusion. The framework not only can test the presence of a particular scenario but also can assign a unique scenario to any observed mortality evolution. Furthermore, it can detect different mortality scenarios for different age ranges in the same population. We also present a methodology for the implementation of our classification framework and apply it to mortality data for U.S. females.

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Notes

  1. All deaths curves in this article are scaled such that the areas underneath the curves each integrate to 1. Thus, the corresponding survival curves start with a radix of 1. Also note that all examples in the second and third sections of this article are based on hypothetical illustrative curves that are, however, reasonable given that overall mortality improves and life expectancy increases.

  2. As noted in the Introduction, the terms expansion, extension, and shifting mortality coexist in the literature. We consider expansion and extension to be the same, and use the term extension for that. We consider shifting mortality to be a different phenomenon, as explained in the next section.

  3. The peak might not be unique in only rather theoretical scenarios—for example, because of multiple peaks of the same height or a plateau. In such a case, one might use a suitable alternative to M or modify the framework to include additional statistics.

  4. In theory, UB can exist only if the probability of death reaches 1 for some age. If the probability of death remains below 1 for all ages, any age could be reached in principle. Research by several authors (see, e.g., Gampe 2010) has indicated that probabilities of death typically flatten out at very old ages, possibly somewhere near 0.5. Thus, the population surviving up to such ages would get halved every year; but if the initial population was large enough, there would be a few survivors up to any age. Therefore, one could argue that UB does not exist in theory, which is, however, irrelevant for our application.

  5. If a distinction between different intensities of increase or decrease is desired, more than three states can be considered or additional information about the slope of the respective trend line (see the section on methodology) can be added.

  6. If the time series has k data points, we consider all k × (k – 1) × (k – 2) / 6 possible triples.

  7. The presentation of the algorithm aims for a clear presentation of and distinction between the steps involved and does not pay attention to computing efficiency.

  8. We also applied the framework to several other populations, such as Sweden, Japan, and West Germany. In all cases, the framework yielded reasonable and informative results. For the sake of brevity, however, we show the results for only one population. We chose U.S. females for illustration because the variety of different observed scenarios was the largest. See Genz (2017) for an application of our framework to a larger number of countries and a comparison of the respective mortality patterns.

  9. We also considered the starting ages 0 (i.e., the complete age range) and 30 in order to exclude effects of young adult’s mortality, such as accidents. The observed scenarios for starting ages 0, 10, and 30 are quite similar.

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Authors and Affiliations

  1. Institut für Finanz- und Aktuarwissenschaften (ifa), Lise-Meitner-Straße 14, 89081, Ulm, Germany

    Matthias Börger, Martin Genz & Jochen Ruß

  2. Institut für Versicherungswissenschaften, Universität Ulm, Ulm, Germany

    Martin Genz & Jochen Ruß

Authors
  1. Matthias Börger
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  2. Martin Genz
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  3. Jochen Ruß
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Correspondence to Martin Genz.

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Börger, M., Genz, M. & Ruß, J. Extension, Compression, and Beyond: A Unique Classification System for Mortality Evolution Patterns. Demography 55, 1343–1361 (2018). https://doi.org/10.1007/s13524-018-0694-3

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  • DOI: https://doi.org/10.1007/s13524-018-0694-3

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