3 ms·

It seems to me that we can get a pretty decent approximation as follows: 1. At any given time, Pr(oldest person <= age x) = Pr(one person <= age x)^N. So (over

by gjm11 14y ago

It seems to me that we can get a pretty decent approximation as follows: 1. At any given time, Pr(oldest person <= age x) = Pr(one person <= age x)^N. So (over time) the median oldest-person age is the (1-2^-1/N) quantile of the age distribution. (For large N, this is roughly 1-log(2)/N.) (You can get that from actuarial tables, or use the Gompertz-Makeham approximation.) 2. So, crudely, the time between oldest-person deaths is comparable to either the expected lifetime of a person of the age found in step 1, or 1/Pr(someone of that age dies in a given year). (Both are approximations. The former will give shorter inter-death times.) 3. According to Wikipedia (which is always right, except when it's wrong), once you get old enough it's a decent approximation to say that that a Very Old Person has about a 50% chance of making it through any given year, and that figure doesn't depend very much on exactly how old they are. Which would suggest that we should get a new oldest person about once every two years, and that for decent-sized populations (say, 1000 or more) the figure should depend only very weakly on population size. If #3 is correct and at very advanced ages the mortality rate is roughly independent of age, it seems like this result shouldn't actually depend much on the details of the probability distributions. (The oldest person alive will almost always be very old.) (You'd get quite different results if, e.g., there were a hard divinely-appointed cutoff at some particular age.)

crntaylor 14y ago
Do you have a link for 3? It would probably affect my answer[0] as I was assuming that the probability of dying between ages x and x+1 increases as a power law after age 60 (which fits well for 60 <= x <= 100, but I don't have data for beyond 100). [0] http://math.stackexchange.com/a/387581/4873 http://math.stackexchange.com/a/387581/4873
- gjm11 14y ago
  All I have for #3 is a brief comment on a Wikipedia page. I wouldn't trust it much, and would in fact expect your model to be nearer the truth. In fact, I'd expect your model to be nearer the truth than it actually is. It seems that the true oldest-person death rate is on the order of 2 per year; my argument yields one per 2 years, your simulation yields one per 1.6 years; is it possible that there's a bug still lurking in your simulation code? (I know you fixed one already! ... and I see you've already responded to my asking the same question over on math.se. [EDIT to add: and I think I may have found a bug, which would make your code underreport oldest-death events. I've reported it on math.se.]) What happens as you twiddle the parameters of your power law to make mortality increase more sharply with increasing age? What do you need to do (if it's possible at all) to get the oldest-person death rate up to 2 per year? It's conceivable that there's some bias in the reporting of deaths of oldest living people that causes too many to be reported -- but I haven't been able to think of any remotely plausible mechanism that would have that effect.
  - crntaylor 14y ago
    The latest iteration of the code (I fixed the bug you pointed out, and another one that I spotted myself) reports 1 death per 0.66 years, which is close to 2 per year. The remaining difference, as you said, could be due to the mortality rate in my model not accelerating fast enough past 100 years old. I'll have a play and let you know.
ademographer 14y ago
As pointed out in 3.) the mortality hazard h(x) does not increase exponentially at very high ages as it does for ages 30 up to let's 80 or 90. Until then the Gompertz-Makeham hazard h(x)= alpha * exp(betax) + gamma approximates adult mortality very well. At higher ages mortality begins to decelerate and, most likely, reaches a plateau at about age 110. References: 1) Parametric models for late life mortality: A useful mortality model, which has a logistic form, is: h(x) = (alpha exp(betax))(1+alpha exp(beta*x)). One can also add an additive term, often denoted as gamma or c. Please see for a comparison for late life models: http://www.demogr.mpg.de/Papers/Books/Monograph5/start.htm http://www.demogr.mpg.de/Papers/Books/Monograph5/start.htm In case you speak German, the German Society of Actuaries has an interesting comparison of models: https://aktuar.de/custom/download/dav/veroeffentlichungen/2004-UAG-Rentnersterblichkeit-DAV-2004R.pdf https://aktuar.de/custom/download/dav/veroeffentlichungen/20... 2) An article which estimates the constant hazard at advanced ages: www.demogr.mpg.de/books/drm/007/3-1.pdf I hope this helps a bit!

by gjm11 14y ago

crntaylor 14y ago

Do you have a link for 3? It would probably affect my answer[0] as I was assuming that the probability of dying between ages x and x+1 increases as a power law after age 60 (which fits well for 60 <= x <= 100, but I don't have data for beyond 100). [0] http://math.stackexchange.com/a/387581/4873 http://math.stackexchange.com/a/387581/4873

gjm11 14y ago
All I have for #3 is a brief comment on a Wikipedia page. I wouldn't trust it much, and would in fact expect your model to be nearer the truth. In fact, I'd expect your model to be nearer the truth than it actually is. It seems that the true oldest-person death rate is on the order of 2 per year; my argument yields one per 2 years, your simulation yields one per 1.6 years; is it possible that there's a bug still lurking in your simulation code? (I know you fixed one already! ... and I see you've already responded to my asking the same question over on math.se. [EDIT to add: and I think I may have found a bug, which would make your code underreport oldest-death events. I've reported it on math.se.]) What happens as you twiddle the parameters of your power law to make mortality increase more sharply with increasing age? What do you need to do (if it's possible at all) to get the oldest-person death rate up to 2 per year? It's conceivable that there's some bias in the reporting of deaths of oldest living people that causes too many to be reported -- but I haven't been able to think of any remotely plausible mechanism that would have that effect.
- crntaylor 14y ago
  The latest iteration of the code (I fixed the bug you pointed out, and another one that I spotted myself) reports 1 death per 0.66 years, which is close to 2 per year. The remaining difference, as you said, could be due to the mortality rate in my model not accelerating fast enough past 100 years old. I'll have a play and let you know.

ademographer 14y ago

As pointed out in 3.) the mortality hazard h(x) does not increase exponentially at very high ages as it does for ages 30 up to let's 80 or 90. Until then the Gompertz-Makeham hazard h(x)= alpha * exp(betax) + gamma approximates adult mortality very well. At higher ages mortality begins to decelerate and, most likely, reaches a plateau at about age 110. References: 1) Parametric models for late life mortality: A useful mortality model, which has a logistic form, is: h(x) = (alpha exp(betax))(1+alpha exp(beta*x)). One can also add an additive term, often denoted as gamma or c. Please see for a comparison for late life models: http://www.demogr.mpg.de/Papers/Books/Monograph5/start.htm http://www.demogr.mpg.de/Papers/Books/Monograph5/start.htm In case you speak German, the German Society of Actuaries has an interesting comparison of models: https://aktuar.de/custom/download/dav/veroeffentlichungen/2004-UAG-Rentnersterblichkeit-DAV-2004R.pdf https://aktuar.de/custom/download/dav/veroeffentlichungen/20... 2) An article which estimates the constant hazard at advanced ages: www.demogr.mpg.de/books/drm/007/3-1.pdf I hope this helps a bit!