

How often does it happen that the oldest person alive dies? - SandB0x
http://math.stackexchange.com/questions/349155/how-often-does-it-happen-that-the-oldest-person-alive-dies

======
StavrosK
I love mathematicians. They give a complicated function, explaining how they
derived each term, and then consider the problem solved, without bothering to
give a number.

It's correct, of course, as the problem _is_ solved, but most people would
attempt a ballpark approximation in the end!

~~~
cschmidt
That is kind of like how you can take a class in number theory, and not
actually use any concrete numbers (like 7, say).

~~~
crntaylor
There's an excellent quotation attributed to Alexandre Grothendieck, one of
the greatest mathematicians alive. At a seminar he was giving on analytic
number theory, someone suggested that they should consider a particular prime
number. "You mean an actual number?" Grothendieck asked. The other person
replied, yes, an actual prime number. Grothendieck replied "All right, take
57."

~~~
JonnieCache
Is there a significance to 57 there?

~~~
lkozma
Yes, it isn't prime, that's the joke (I guess).

~~~
Someone
Yes, but not in the sense that Grothendieck made a joke by saying '57', but in
the sense that Grothendieck, who knows quite a bit about prime numbers, said
'57' without realizing that it is composite.

~~~
lkozma
Yes, I understood it in this sense too :) Quite a character, Grothendieck.

------
gjm11
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
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>

~~~
gjm11
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
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.

------
tokenadult
The person who asked the question on math.stackexchange.com referred to news
reports, and is asking what is essentially a historical question, so the
question really should have been asked on a question-and-answer site about
historical research rather than on a site about mathematics. That's why the
answers are so irrelevant to the nature of the question.

The Nexis commercial database of news stories may be comprehensive enough
these days to answer a question like that in detail going back to your own
birth year. It would cost money to do the Nexis search, and you'd probably
have to pay someone to pore through the search results and edit a document
that would accurately summarize the results, but this should be a solvable
problem these days.

(As another comment here has already pointed out, the basic answer is "Every
time someone becomes the oldest person in the world, that person eventually
dies," but I take it that the question actually asked means "How often does
the identity of the 'oldest person in the world' change to being a new
individual?")

~~~
gwern
> That's why the answers are so irrelevant to the nature of the question.

I posted a historical response there, so hopefully that settles the issue.

~~~
tokenadult
Thank you for that. I see the French woman Jeanne Louise Calment (21 February
1875 – 4 August 1997) who long held the title of world's oldest person is an
impressive outlier.

I heard a description of the life of Calment by a local researcher who
participates in longitudinal studies of extreme old age. She outlived her
husband, and all her (few) descendants. She only gave up smoking when she
became so blind that she could no longer see the end of a cigarette to light
it. The researcher told me this story (taken here from Wikipedia) about how
she kept living in her house after she became aged and widowed:

"In 1965, aged 90 years and with no heirs, Calment signed a deal to sell her
former apartment to lawyer André-François Raffray, on a contingency contract.
Raffray, then aged 47 years, agreed to pay her a monthly sum of 2,500 francs
until she died. Raffray ended up paying Calment the equivalent of more than
$180,000, which was more than double the apartment's value. After Raffray's
death from cancer at the age of 77, in 1995, his widow continued the payments
until Calment's death."

------
kyllo
You can't trust the actual data on this one, because the "oldest person alive"
is often a deceased Japanese person whose death has not been reported, for the
purpose of pension fraud.

~~~
gwern
That's why they start by mentioning the GRG, which does do some investigation
of its official entries and is aware of issues like pension fraud.

~~~
gwern
If anyone is curious, I just updated the post with a quickie analysis of the
GRG data (I'm on the mailing list and brought this and the post up and someone
else posted the relevant table). Turns out the actual timing differs on
whether you're looking at mean or median, because Calment screwed things up by
living a ridiculously long time.

------
jeremysmyth
Every time. 100%. Next question please!

------
pierrebai
Since the question is about the average, it seems the question can be simply
answered: the next-oldest person, on average, will die at the same age as the
current oldest person. (Obviously, assuming an unchanging mortality rate over
time, but this sounds like a valid approximation for very old people. There
seem to be a wall around 122.) Thus the average waiting time between two
oldest-person deaths is the average age difference between the two oldest
living persons.

Edit: actually, an even simpler first-order approximation is possible. If we
take at face-value that very old people have a 50% chance of living one more
year, and that this statistics holds whatever the baseline date, then upon the
death of the eldest person, the average life-span of the next eldest person is
1/2 + 1/4 + 1/8 ... IOW, 1 year.

------
JohnLBevan
Should it be of interest, here's a list of those people born in 1800s still
alive today (i.e. over 113 years old):
[http://en.wikipedia.org/wiki/Last_surviving_1800s-born_peopl...](http://en.wikipedia.org/wiki/Last_surviving_1800s-born_people)

~~~
JohnLBevan
ps. All are from Japan, US, Italy or UK. I suspect that may be down to record
keeping as much as lifestyle. For example, a friend's wife is from Turkey and
doesn't know how old she is as her date of birth was never recorded; one year
her parents just made a guess saying "well, you were born in summer and you
look like an 8 year old, so we'll stick you down as 21st June 1965".

------
jaynos
This seems like one of those interview questions where there is no right or
wrong answer, they just want to see your method. I'll probably waste most of
my day thinking about this.

------
sageikosa
For the last 20 oldest, the five person rolling length of time as oldest seems
to be hanging around 200 days.

------
Kiro
I need a number.

~~~
uptown
3

~~~
Kiro
So the oldest person dies every 3 years.

------
ExpiredLink
tq;dr

tq = teaser question

------
ttrreeww
Depends, in some specie, the older you are, the less likely you are to die.

