Theoretical Population Biology 80 (2011) 64–70 Contents lists available at ScienceDirect Theoretical Population Biology journal homepage: www.elsevier.com/locate/tpb Admissible mixing distributions for a general class of mixture survival models with known asymptotics Trifon I. Missov a,b,∗ , Maxim Finkelstein c,a a Max Planck Institute for Demographic Research, Konrad-Zuse-Str. 1, 18057 Rostock, Germany b Institute of Sociology and Demography, University of Rostock, Ulmenstr. 69, 18051 Rostock, Germany c Department of Mathematical Statistics, University of the Free State, P.O. Box 339, 9300 Bloemfontein, South Africa article info Article history: Received 8 November 2010 Available online 10 May 2011 Keywords: Mortality asymptotics Mixture survival model Frailty distribution Function of regular variation Tauberian theorem abstract Statistical analysis of data on the longest living humans leaves room for speculation whether the human force of mortality is actually leveling off. Based on this uncertainty, we study a mixture failure model, introduced by Finkelstein and Esaulova (2006) that generalizes, among others, the proportional hazards and accelerated failure time models. In this paper we first, extend the Abelian theorem of these authors to mixing distributions, whose densities are functions of regular variation. In addition, taking into account the asymptotic behavior of the mixture hazard rate prescribed by this Abelian theorem, we prove three Tauberian-type theorems that describe the class of admissible mixing distributions. We illustrate our findings with examples of popular mixing distributions that are used to model unobserved heterogeneity. © 2011 Elsevier Inc. All rights reserved. 1. Introduction Mortality in the developed countries decreased steadily in the second half of the twentieth century (Tuljapurkar et al., 2000). This lead to the emergence of a growing number of individuals, the so-called supercentenarians, surviving to ages above 110. The availability of data at increasingly higher ages facilitates the meaningful statistical estimation of the human force of mortality further and further along the age axis. The latter brings additional insight into the choice of adequate theoretical mortality models. In the classical framework the individual hazard of death follows a baseline schedule, modulated in a certain functional form by a realization of a random variable, called frailty, accounting for unobserved heterogeneity, whereas the population force of mortality is a mixture of the baseline mortality and the frailty distribution. The choice of these two distributions, as well as the choice of a functional form that captures the effect of frailty on individual mortality, determines the asymptotic behavior of the population hazard of death, i.e. its behavior at the highest ages. The International Database of Longevity (IDL, 2010) offers de- tailed information on thoroughly validated cases of supercentenar- ians. Using these data, Gampe (2010) estimated the human force of mortality after age 110. Her analysis revealed that human mortality ∗ Corresponding author at: Max Planck Institute for Demographic Research, Konrad-Zuse-Str. 1, 18057 Rostock, Germany. E-mail addresses: Missov@demogr.mpg.de (T.I. Missov), FinkelM.SCI@mail.uovs.ac.za (M. Finkelstein). between ages 110 and 114 levels off regardless of gender. It is flat at a level corresponding approximately to a 50% annual probabil- ity of death (Gampe, 2010; Robine et al., 2005). As human popula- tions are heterogeneous, this finding raises an important question, which this article addresses: what is the underlying heterogeneity model and how is individual frailty distributed if (i) human mor- tality approaches a constant limit, or (ii) mortality abandons the plateau at later ages, where there are still no officially recorded sur- vivors, i.e. if the asymptotic behavior of the force of mortality is not constant? We address this problem in much more general settings, not restricting ourselves to a Gompertz baseline distribution and an asymptotically flat force of mortality. In fact, for a rather gen- eral frailty survival model (which includes as special cases propor- tional hazards and accelerated life models) and given asymptotic behavior of the hazard rate (e.g., flat force of mortality at infinity), we describe the class of admissible frailty distributions that ‘‘gen- erates’’ this behavior. A similar inverse problem was studied by Steinsaltz and Wachter (2006). It was restricted, though, to the special case of the proportional hazards frailty model. Assuming that the baseline hazard is asymptotically equivalent to a Gompertz curve and the frailty (mixing) distribution behaves in a neighborhood of zero like a power function cz α , where c ≡ const and α> −1, these authors prove an Abelian theorem that the resulting mixture (population) hazard rate is asymptotically flat. Finkelstein and Esaulova (2006) assume the same behavior of the frailty distribution for z → 0, but for a more general survival model. They derive independently the asymptotic result of Steinsaltz and Wachter (2006) and, moreover, prove that the mixture hazard rate for the accelerated life model 0040-5809/$ – see front matter © 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.tpb.2011.05.001