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The Biostatistics of Aging: From Gompertzian Mortality to an Index of Aging-Relatedness by Gilberto Levy, Bruce Levin

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4THE INDEX OF AGING-RELATEDNESS

In previous chapters, we laid out an extensive theoretical foundation for the index of aging-relatedness proposed here, which we briefly recapitulate. The Gompertz distribution was developed based on empirical human mortality observations and has played a central role in demography for describing the survival time of human populations. The Gumbel-type distribution is one of just two possible limiting distributions of the minimum of i.i.d. time-to-event random variables and becomes the Gompertz distribution when restricted to the positive half line. The other limiting distribution of the minimum of i.i.d. time-to-event random variables is the Weibull-type distribution, which plays a prominent role in reliability theory for describing the time to failure of mechanical devices. In the context of offering an account of the Gompertz pattern of mortality through the statistical theory of extreme values and biological evolution, we extended the applicability of some classical theorems on the asymptotic behavior of the minimum of time-to-event random variables. These results showed that if the initial distributions belong to a certain family of functions that are infinitely flat at the lower endpoint of the distribution (i.e., all derivatives evaluated at the lower endpoint of the distribution equal 0), the limiting distribution of the minimum is the Gumbel-type distribution; and if the initial distributions belong to a family of functions that are regularly ...

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