Abstract
In this review paper formulas for survival functions are derived that take into account risks of deaths in early life including infancy, mid life, a random component of deaths due to accidents and deaths in older ages. The basic ideas used in the derivation of survival function for each of the components just mentioned are risk functions. Given a formula for a risk function, it is possible to derive a formula for the corresponding survival function. By using the theory of competing risks, a formula for survival function that takes into account the risks of deaths in various stages of life expressed as a product of survival functions for the risks of deaths under consideration. For many applications information on the numerical values of parameters in survival functions is not available. Consequently, rationales are developed for assigning plausible values to parameters that take into account personal ideas of an investigator may have for each stage of life. For every assignment of parameter values in the paper, a numerical version of survival functions are plotted in graphs so that an assessment of the plausibility of the chosen parameter values may be made. Also included in the paper is an application of survival functions in an experiment to make an assessment as to whether a small population of chimpanzees, or some other endangered species of animals, will have descendants that make up a surviving population 200 years into the future.
Highlights
In computer simulation experiments designed to study the evolution of age structured animal or human populations with respect to two or more genetic autosomal loci in deep time, such as hundreds or thousands of years before the present, it is almost always the case that no mortality data exists
By using the theory of competing risks, a formula for survival function that takes into account the risks of deaths in various stages of life expressed as a product of survival functions for the risks of deaths under consideration
For further details on the proposals of Gompertz and Makeham, the book by Smith and Keyfitz The work of Thiele is presented In the book and the formula that Thiele proposed is presented on page 72
Summary
In computer simulation experiments designed to study the evolution of age structured animal or human populations with respect to two or more genetic autosomal loci in deep time, such as hundreds or thousands of years before the present, it is almost always the case that no mortality data exists. The formula for the parametric function used in the two papers for the same three age intervals under consideration were different, the same method of splicing the survival functions together was the same for both papers, which provided a parametric survival function for the entire age interval [0, k] Both models fit survival functions estimated from many period life tables, they are mildly flawed, because of small discontinuities at the splice points. A four parameter parametric survival function that was used to produce the content of the paper was based on partitioning the life span of each individual into two age intervals [0, 30) and [30, r) For this model, age x = 30 was a splice point and there was a noticeable discontinuity at that age point that appeared in age distribution estimated from data generated in Monte Carlo simulation experiments as well as those computed using the embedded deterministic model. In the remaining sections of this paper, these potential flaws will be investigated and alternative procedures will be investigated
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