Symmetries between life lived and left in finite stationary populations

Abstract

(ProQuest: ... denotes formulae omitted.)1. IntroductionThe symmetries between life lived and life left in stationary populations have drawn the attention of several scholars in past years. Examples are found in the fields of demography and population biology (Goldstein 2009, 2012; Kim and Aron 1989; Muller et al. 2004, 2007; Riffe 2015; Vaupel 2009), and reliability engineering (Cha and Finkelstein 2016; Finkelstein and Vaupel 2015). One of the main results that has recently emerged is equality, as first coined by Vaupel (2009), which establishes that the age composition equals the distribution of remaining lifetimes in stationary populations. To our knowledge, this relationship was first described by Brouard (1989), and was later and independently noticed by James Carey in the study of the survival patterns of captive and follow-up cohorts of med-flies (Muller et al. 2004, 2007). Hence, we choose to refer to this relationship as the 'Brouard-Carey equality.'3Vaupel (2009) proved that in stationary populations of infinite size and continuous time, the probability that a randomly selected individual is age x equals the probability that the individual has exactly x time left until death. Formally,(1) ...where the probability density function c(x) describes the age composition of the population, and the probability density function g(y) gives the distribution of remaining lifetimes. This result has several applications in the study of human and non-human populations with unknown ages. For instance, in capture-recapture studies in which are captured and then followed until death, assuming strict and deterministic stationarity and the absence of other biases, it can be inferred that the unobserved distribution of ages at capture is equal to the observed distribution of the follow-up durations.Rao and Carey (2015) claim to have an alternative and innovative proof of the Brouard-Carey equality that does not depend on any of the classical works on renewal equation and renewal theory, and that is inspired by experimental observations of captive (Rao and Carey 2015: 587-588). Hence, their approach is an attempt to prove the Brouard-Carey equality for empirical and finite stationary populations, rather than theoretical infinite populations, as is the case in Vaupel (2009). However, the use of continuous time in finite populations represents a subtle flaw in Rao and Carey's work: In order to validate the Brouard-Carey equality for finite stationary populations, time measures need to be explicitly discretized.In the following section we analyze Theorem 1 by Rao and Carey (2015). Next, we discuss why the symmetries between life lived and life left in finite stationary populations are not valid under continuous time, and that the adoption of a discrete-time framework is imperative. Finally, we suggest a reformulation of the Brouard-Carey equality for finite stationary populations.2. Rao and Carey's Theorem 1The motivation underlying Rao and Carey's approach to prove the Brouard-Carey equality is the observation of survival patterns in captive med-flies. As stated by the authors, their theory and method of proof uses sequentially arranged data of captive individuals (Rao and Carey 2015: 588), which implies empirical - and consequently finite - stationary populations.If the Brouard-Carey equality were true for finite stationary populations, one would expect the number of of any age x to equal the number of with x life left in the same population at any time point. Rao and Carey (2015) adopt a slightly different approach, suggesting the existence of a graphical symmetry between the sets of capture ages and follow-up durations of a finite stationary population.4 The following is the statement of Theorem 1 by Rao and Carey (2015: 584-585):Theorem 1 (Rao and Carey). Suppose (X, Y, Z) is a triplet of column vectors, where X = [x1, x2,. …