Background: The gamma-Gompertz multiplicative frailty model is the most common parametric model applied to human mortality data at adult and old ages. The resulting life expectancy has been calculated so far only numerically. Objective:... more
Background: The gamma-Gompertz multiplicative frailty model is the most common parametric model applied to human mortality data at adult and old ages. The resulting life expectancy has been calculated so far only numerically. Objective: Properties of the gamma-Gompertz distribution have not been thoroughly studied. The focus of the paper is to shed light onto its first moment or, demographically speaking, characterize life expectancy resulting from a gamma-Gompertz force of mortality. The paper provides an exact formula for gamma-Gompertz life expectancy at birth and a simpler high-accuracy approximation that can be used in practice for computational convenience. In addition, the article compares actual (life-table) to model-based (gamma-Gompertz) life expectancy to assess on aggregate how many years of life expectancy are not captured (or overestimated) by the gamma-Gompertz mortality mechanism. Comments: A closed-form expression for gamma-Gomeprtz life expectancy at birth contains a special (the hypergeometric) function. It aids assessing the impact of gamma-Gompertz parameters on life expectancy values. The paper shows that a high-accuracy approximation can be constructed by assuming an integer value for the shape parameter of the gamma distribution. A historical comparison between model-based and actual life expectancy for Swedish females reveals a gap that is decreasing to around 2 years from 1950 onwards. Looking at remaining life expectancies at ages 30 and 50, we see this gap almost disappearing.
BACKGROUND The Gompertz force of mortality (hazard function) is usually expressed in terms of a, the initial level of mortality, and b, the rate at which mortality increases with age. OBJECTIVE We express the Gompertz force of mortality... more
BACKGROUND The Gompertz force of mortality (hazard function) is usually expressed in terms of a, the initial level of mortality, and b, the rate at which mortality increases with age. OBJECTIVE We express the Gompertz force of mortality in terms of b and the old-age modal age at death M and present similar relationships for other widely-used mortality models. Our objective is to explain the advantages of using the parameterization in terms of M. METHODS Using relationships among life table functions at the modal age at death, we derive the Gompertz force of mortality as a function of the old-age mode. We estimate the correlation between the estimators of old (a and b) and new (M and b) parameters from simulated data. RESULTS When the Gompertz parameters are statistically estimated from simulated data, the correlation between estimated values of b and M is much less than the correlation between a and b. For the populations in the Human Mortality Database there is a negative association between a and b and a positive association between M and b. CONCLUSIONS Using M, the old-age mode, instead of a, the level of mortality at the starting age, has two major advantages. First, statistical estimation is facilitated by the lower correlation between the estimators of model parameters. Second, estimated values of M are more easily comprehended and interpreted than estimated values of a.
The article aims at describing 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... more
The article aims at describing 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) 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. 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.
The resampling method was applied to generate the statistical distribution of TFR of Bolivia in 1998 and 2003 so as to evaluate differences in the estimator at both times. Samples of ENDSAS for 1998 and 2003 of Bolivia, which have a... more
The resampling method was applied to generate the statistical distribution of TFR of Bolivia in 1998 and 2003 so as to evaluate differences in the estimator at both times. Samples of ENDSAS for 1998 and 2003 of Bolivia, which have a stratified and two-stage design, were considered. The distributions turnout to be positive for the normality tests; nonetheless, they have their own characteristics of asymmetry and kurtosis. The method allows generating the empirical statistical distribution for the estimator, which is not feasible in the reality. It also allows evaluating the precision of the estimator as the replication algorithm can be modified and applied to other populations or other indicators.
In a Gompertz mortality model with constant yearly improvements at all ages, linear increases in period life expectancy correspond to linear increases in the respective cohort life expectancy. The link between the two measures can be... more
In a Gompertz mortality model with constant yearly improvements at all ages, linear increases in period life expectancy correspond to linear increases in the respective cohort life expectancy. The link between the two measures can be given by a simple approximate relationship.
