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I want to simulate more or less realistic survival times for humans. I want to use a basic distribution$^*$ to do so. What distribution and what parameters would, to some extent, resemble the survival of humans$^{**}$?

$^*$ For instance, a Weibull with a certain combination of parameters.

$^{**}$Assuming an ideal scenario: no wars, no epidemics, etc.

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  • $\begingroup$ Welcome to CV. By "human survival times," are you referring to mortality tables? $\endgroup$
    – user78229
    Jul 7, 2016 at 16:39
  • $\begingroup$ @DJohnson hmm, good question. I was just thinking of the distribution of the age at dead of humans (assuming all kinds of homogeneity), but I think you make a good point and I should have a look at mortality tables. $\endgroup$
    – July
    Jul 7, 2016 at 16:45
  • $\begingroup$ Do you mean "assuming all kinds of heterogenetity?" One source for mortality data in the US is the CDC's NVSS... cdc.gov/nchs/nvss/deaths.htm $\endgroup$
    – user78229
    Jul 7, 2016 at 18:39

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You can extract the distribution of the age at death from mortality tables or from life tables (which are often publicly available) via simple calculation. A good source of such information across many countries is the Human Mortality Database (for accessing this you need to register and be approved, but the data are free). However, in many countries life table data is often also available from government sources or similar agencies, in some cases online.

For example, I got the data for this plot (NZ male death probabilities, 2005-2007):

enter image description here

from here (in the sidebar box labelled, "Downloads", second link).

[However, these life-table distributions will represent a mixture across cohorts, so they won't actually represent the distribution for any given cohort. They're also always a at least few years out of date, unless you're working with projections. Nevertheless they're about as good as you're likely to obtain easily.]

These do not look like any simple common distribution you'd find in a textbook, and vary from country to country as well as over time. There's usually a spike at the very youngest ages and a big broad hump at the older ages. While these distributions are given for discrete ages, with there being typically around 100 ages under consideration, you may well consider using continuous distributions to approximate these.

For a first world country with good health-care, you tend to see a relatively small spike at infancy and a big mass, left skew but not too far from symmetric (much as we see above), peaking in the vicinity of 80 or so (earlier if you go back in time, later in countries with good health care of the broader population; the peak above is at 87) but with a wide spread about it. Further examples here and here.

If you want to use "standard" distributions you'll need to use a mixture of them. You might do more or less okay with a two- to four- component mixture distribution for these cases. For countries with long life expectancy and low young-age mortality the broad peak might be treated as roughly normal, but you'll probably want a third component to represent mortality at the middle ages.

[Such curves - characterized by low infant mortality - may also be found in poorer countries which nevertheless have relatively good healthcare systems, and conversely higher infant mortality may be seen in otherwise wealthy countries that have a strong division between rich and poor and don't have good health coverage of the poor.]

Countries with lower life-expectancy tend to have much higher infant mortality (with the peak at the youngest ages being far higher, and extending out several years) and much more of a left tail on the broad peak (as well as the older peak being downshifted somewhat).

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For a review of models which have been used you might want to look at

@ARTICLE{thatcher99,
  author = {Thatcher, A R},
  year = 1999,
  title = {The long-term pattern of adult mortality and the highest attained
          age},
  journal = {Journal of the Royal Statistical Society, Series A},
  volume = 162,
  pages = {5--43}
}

There is also a substantial literature on the extremes of human life span. Although this is not what you asked it is a cautionary tale about the difficulties of modelling in this field. A possible first read is

@ARTICLE{olshansky90,
  author = {Olshansky, S J and Carnes, B A and Cassel, C},
  title = {In search of {M}ethuselah: estimating the upper limits to human
          longevity},
  year = 1990,
  journal = {Science},
  volume = 250,
  pages = {634--640}
}
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