Trifon I. Missov

Admissible mixing distributions for a general class of mixture survival models with known asymptotics

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.

Mortality Implications of Mortality Plateaus

This article aims to describe in a unified framework all plateau-generating random effects models in terms of (i) plausible distributions for the hazard (baseline mortality) and the random effect (unobserved heterogeneity, frailty) as well as (ii) the impact of frailty on the baseline hazard. Mortality plateaus result from multiplicative (proportional) and additive hazards, but not from accelerated failure time models. Frailty can have any distribution with regularly-varying-at-0 density and the distribution of frailty among survivors to each subsequent age converges to a gamma distribution. In a multiplicative setting the baseline cumulative hazard can be represented as the inverse of the negative logarithm of any completely monotone function. If the plateau is reached, the only meaningful solution at the plateau is provided by the gamma-Gompertz model.

Goodness-of-fit tests for the Gompertz distribution

Abstract While the Gompertz distribution is often fitted to lifespan data, testing whether the fit satisfies theoretical criteria is being neglected. Here four goodness-of-fit measures – the Anderson–Darling statistic, the correlation coefficient test, a statistic using moments, and a nested test against the generalized extreme value distributions – are discussed. Along with an application to laboratory rat data, critical values calculated by the empirical distribution of the test statistics are also presented.

Quantifying the Shape of Aging

In Biodemography, aging is typically measured and compared based on aging rates. We argue that this approach may be misleading, because it confounds the time aspect with the mere change aspect of aging. To disentangle these aspects, here we utilize a time-standardized framework and, instead of aging rates, suggest the shape of aging as a novel and valuable alternative concept for comparative aging research. The concept of shape captures the direction and degree of change in the force of mortality over age, which—on a demographic level—reflects aging. We 1) provide a list of shape properties that are desirable from a theoretical perspective, 2) suggest several demographically meaningful and non-parametric candidate measures to quantify shape, and 3) evaluate performance of these measures based on the list of properties as well as based on an illustrative analysis of a simple dataset. The shape measures suggested here aim to provide a general means to classify aging patterns independent of any particular mortality model and independent of any species-specific time-scale. Thereby they support systematic comparative aging research across different species or between populations of the same species under different conditions and constitute an extension of the toolbox available to comparative research in Biodemography.

Stochasticity, heterogeneity, and variance in longevity in human populations

Inter-individual variance in longevity (or any other demographic outcome) may arise from heterogeneity or from individual stochasticity. Heterogeneity refers to differences among individuals in the demographic rates experienced at a given age or stage. Stochasticity refers to variation due to the random outcome of demographic rates applied to individuals with the same properties. The variance due to individual stochasticity can be calculated from a Markov chain description of the life cycle. The variance due to heterogeneity can be calculated from a multistate model that incorporates the heterogeneity. We show how to use this approach to decompose the variance in longevity into contributions from stochasticity and heterogeneous frailty for male and female cohorts from Sweden (1751–1899), France (1816–1903), and Italy (1872–1899), and also for a selection of period data for the same countries. Heterogeneity in mortality is described by the gamma-Gompertz–Makeham model, in which a gamma distributed “frailty” modifies a baseline Gompertz–Makeham mortality schedule. Model parameters were estimated by maximum likelihood for a range of starting ages. The estimates were used to construct an age×frailty-classified matrix model, from which we compute the variance of longevity and its components due to heterogeneous frailty and to individual stochasticity. The estimated fraction of the variance in longevity due to heterogeneous frailty (averaged over time) is less than 10% for all countries and for both sexes. These results suggest that most of the variance in human longevity arises from stochasticity, rather than from heterogeneous frailty.

Gompertz-Makeham life expectancies: expressions and applications.

In a population of individuals, whose mortality is governed by a Gompertz-Makeham hazard, we derive closed-form solutions to the life-expectancy integral, corresponding to the cases of homogeneous and gamma-heterogeneous populations, as well as in the presence/absence of the Makeham term. Derived expressions contain special functions that aid constructing high-accuracy approximations, which can be used to study the elasticity of life expectancy with respect to model parameters. Knowledge of Gompertz-Makeham life expectancies aids constructing life-table exposures.

Integral Evaluation Using the Δ2-distribution. Simulation and Illustration

The Δ2-distribution is a multivariate distribution, which plays an important role in variance reduction of Monte Carlo integral evaluation. Selecting the nodes of random cubature formulae according to Δ2 ensures an unbiased and efficient estimate of the studied integral regardless of the region it is solved over. The Δ2 distribution is also relevant in problems such as separating errors in regression analysis and constructing D-optimal designs in multidimensional regions. Inefficient simulation of Δ2 prevented the application of the underlying theory in real problems. Ermakov and Missov [S.M. Ermakov and T.I. Missov, On Simulation of the Δ2-distribution. Vestnik St. Petersburg University, issue 4 (2005), 123–140.], proposed an algorithm which combines all rejection, inversion, and mixture techniques. Its complexity allows simulating Δ2 vectors of big lengths. Moreover, it works in the most general settings of the problem of integral evaluation. This article presents a modification of the simulation algorithm as well as its illustration for a popular integral in Reliability Theory.

Revisiting Mortality Deceleration Patterns in a Gamma-Gompertz-Makeham Framework

We calculate life-table aging rates (LARs) for overall mortality by estimating a gamma-Gompertz-Makeham (ΓGM) model and taking advantage of LAR’s parametric representation by Vaupel and Zhang (Demogr Res 23(26), 737–748, 2010). For selected HMD countries, we study how the evolution of estimated LAR patterns could explain observed (1) longevity dynamics, and (2) mortality improvement or deterioration at different ages. Surprisingly, the age of mortality deceleration x∗ showed almost no correlation with a number of longevity measures apart from e0. In addition, as mortality concentrates at older ages with time, its characteristic bell-shaped pattern becomes more pronounced. Moreover, in a ΓGM framework, we identify the impact of senescent mortality on shape of the rate of population aging. We also find evidence for a strong relationship between x∗ and the statistically significant curvilinear changes in the evolution of e0 over time. Finally, model-based LARs appear to be consistent with point (b) of the “heterogeneity hypothesis” (Horiuchi and Wilmoth, J Gerontol Biol Sci 52A(1), B67–B77, 1997): mortality deceleration, due to selection effects, should shift to older ages as the level of total adult mortality declines.

On importance sampling in the problem of global optimization

Abstract Importance sampling is a standard variance reduction tool in Monte Carlo integral evaluation. It postulates estimating the integrand just in the areas where it takes big values. It turns out this idea can be also applied to multivariate optimization problems if the objective function is non-negative. We can normalize it to a density function, and if we are able to simulate the resulting p.d.f., we can assess the maximum of the objective function from the respective sample.

Adequate life-expectancy reconstruction for adult human mortality data

Mortality information of populations is aggregated in life tables that serve as a basis for calculation of life expectancy and various life disparity measures. Conventional life-table methods address right-censoring inadequately by assuming a constant hazard in the last open-ended age group. As a result, life expectancy can be substantially distorted, especially in the case when the last age group in a life table contains a large proportion of the population. Previous research suggests addressing censoring in a gamma-Gompertz-Makeham model setting as this framework incorporates all major features of adult mortality. In this article, we quantify the difference between gamma-Gompertz-Makeham life expectancy values and those published in the largest publicly available high-quality life-table databases for human populations, drawing attention to populations for which life expectancy values should be reconsidered. We also advocate the use of gamma-Gompertz-Makeham life expectancy for three reasons. First, model-based life-expectancy calculation successfully handles the problem of data quality or availability, resulting in severe censoring due to the unification of a substantial number of deaths in the last open-end age group. Second, model-based life expectancies are preferable in the case of data scarcity, i.e. when data contain numerous age groups with zero death counts: here, we provide an example of hunter-gatherer populations. Third, gamma-Gompertz-Makeham-based life expectancy values are almost identical to the ones provided by the major high-quality human mortality databases that use more complicated procedures. Applying a gamma-Gompertz-Makeham model to adult mortality data can be used to revise life-expectancy trends for historical populations that usually serve as input for mortality forecasts.