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Today, we are brought the sad news that Europe's oldest woman died. A little over a week ago the oldest person in the U.S. unfortunately died. Yesterday, the Netherlands' oldest man died peacefully. The Gerontology Research Group keeps records: Guinness World Records.

If you live in a country with $N_{\text{country}}$ people, a continent with $N_{\text{continent}}$ people, and a world with $N_{\text{world}}$ people, during a year and on average, how often will you be notified (if you're paying attention to your quality tabloid) of the death of the oldest man/woman/person alive of your country/continent/world? (Note that a death will result in at most one notification.)


Edit, Suggestions (due to comments; thanks!) only:

Ultimately, I am looking for a realistic formula. That means that life tables are certainly allowed (but mind the ending, and note that some of the oldest people are older than the maximum in the tables).

I guess that the Gompertz–Makeham law of mortality, or any plausible model of late-life mortality, is also fair game.

However, these suggestions do not preclude some insight or realistic assumption that might not need any of those things.

Further potentially useful assumptions (with regard to a subpopulation) may include (if reasonable w.r.t. the question):

  1. Time is discrete.
  2. The number of births at any time is constant.
  3. The number of deaths at any time is constant.
  4. The number of births is equal to the number of deaths at any time.
  5. Time is defined in such a way that, at any time, one birth and one death occur.
  6. In the limit to infinite age, the probability of a person of such age dying goes to $1$.
  7. At any time, the oldest person(s) alive is (are) the most likely to die.
  8. At any time, an older person(s) alive is (are) more likely to die than any younger person alive.

(Although such assumptions aren't necessarily realistic, I don't immediately see how they would distort the outcome.)

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Don't you need more information than this? "everyone lives forever" satisfies the criteria in your question, and then you're never notified. If this were a mayfly population there would be multiple notices in a minute. What are we taking as the average lifespan? –  Robert Mastragostino Apr 2 '13 at 15:05
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@Ilya: Slightly morbid, perhaps, but I don't see how it's unethical. I think it's a very interesting mathematical question and I look forward to an answer. The title is a little confusing as it looks like a news item rather than a question; I'm going to edit it. –  Nate Eldredge Apr 2 '13 at 15:06
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@Nate: perhaps, you're right - it's a subjective opinion only in the end. The question clearly is of interest to the actuarial science. Just to clarify: didn't downvote, didn't vote to close, I think community decides. And thanks for fixing the title - it was less appropriate before. –  Ilya Apr 2 '13 at 15:10
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An extremely oversimplified but perhaps tractable version of this question: Consider $N$ independent Poisson processes with the same parameter (which may be taken to be $1$). The “age” of process $i$ at any given time is the time since it last fired. How often will the oldest process fire? My own intuition says one in $N$ events will involve the oldest process, since it is a defining property of Poisson processes that their future is independent of their past. This example is in a sense the opposite extreme from what @joriki suggests; I conjecture that the general answer is between these two. –  Harald Hanche-Olsen Apr 2 '13 at 15:25
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To the title question: exactly once. –  Will Jagy Apr 2 '13 at 18:42

9 Answers 9

An approach would be to assume:

  • a non-changing (period) life table and hence
  • a distribution of the age of a random person.

Let this distribution be called $F$, so the density of the oldest person of a population of $N$ being $t$ years should be $\frac{d}{dt}F(t)^N$. With the constant life table one can calculate the conditional density of having to wait $t'$ until the oldest person dies, given the oldest person is now $t$ years old. Let this density be called $g$. With this the expected time until an oldest person dies would be:

$\int_0^\omega \frac{d}{dt}F(t)^N \int_t^\omega g(t') \cdot t' dt' dt$

If one does not want to impose an ultimate age $\omega$ one can set it to infinity. Perhaps this does not quite answer your question as it is the expected waiting time of any given moment and not conditioned on the oldest person just having died, but it might point others to the right solution.

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This problem is similar to finding the distribution of a stochastic process at a certain time. In that case you need to solve a PDE for which you need some initial condition in order to find a unique solution.

So, the solution of this problem involves the knowledge of the distribution of ages for the living people in a given population at time $t_0$. This means that we know at this time how many people of age 1, age 2, age 3 ..., age 122,... we have in our population.

Notation : ${}_xP_n$ -probability of a a person of age $n$ to survive $x$ years(lives up to $x+n$); ${}_xQ_n$ -probability of a a person of age $n$ to die in the next $x$ years(dies before $x+n$); $E_n$ -expectation of life of a a person of age $n$ (how many years will she live on average); $l_n$ - number of people having age $n$ at time $t_0$; $h$ -highest life for which ${}_1P_h$ is positive.

Now, the person $Y$ born at time $t_0$ is the youngest person alive. The probability that this person to be the oldest person alive at some time $t_x$ (where $t_x=t_0+x$) in the future can be calculated as the product of: the probability that $Y$ lives up to the age $x$ and the probabilities that the people having ages $n_1,n_2,...$ at time $t_0$ live less than (die before) $x+n_1,x+n_2,...$: \begin{equation} O_x={}_xP_0 \prod_{n=1}^h{}_{x+n}Q_0^{l_n } \end{equation} We compute $O_x$ for $x\in\{1,2,\dots,h \}$ to obtain the probabilities that $Y$ is the oldest person for various values of $x$. Now, at a general time $t_x$, the person $Y$ is the oldest person with probability $O_x$ and her life expectancy is $E_x$. Given the formula for the life expectancy $E_x=\sum_{t=1}^h {}_tP_x $ we can say that, on average, the oldest person at a general time $t_x$ will live: \begin{equation} E_x \cdot O_x=\sum_{t=1}^h {}_tP_x \cdot \Big[{}_xP_0 \prod_{n=1}^h{}_{x+n}Q_0^{l_n }\Big] \end{equation}

Note that this solution doesn't guarantee that at time $t_x$, the person $Y$ is the only person of age $x$ (not the unique oldest person). To take the uniqueness into consideration we will have to multiply the relation above by the probability that the other people born at $t_0$ die before $t_x$.

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Super-simple answer, depending only on some crude properties of mortality rates:

It appears that at very advanced ages, mortality rate varies only slowly with age and is on the order of 50% per year. (If this is wrong, everything else in this answer is wrong. On the other hand, if it's right then it's all we need.)

It will basically always be the case that the oldest person alive is very old.

Therefore, at any given time the death-of-oldest-person process is approximately Poisson with rate 1/2, so typical gaps between these events will be on the order of 2 years.

[EDITED to add: A bit of wikipedia-grovelling suggests that recent oldest-person deaths have been: Dina Manfredini, 2012-12; Besse Cooper, 2012-12; Maria Gomes Valentim, 2011-06; Eunice Sanborn, 2011-01; Eugenie Blanchard, 2010-10. That's a lot more frequent, which suggests that mortality accelerates pretty rapidly at extreme ages. I'm puzzled because Chris Taylor's simulation gives results comparable to mine despite having that feature, even after he fixed his bug; perhaps he has another? :-)]


This is only weakly sensitive to population size. (In a population of size N, the median age of the oldest person will be roughly the 1-log(2)/N quantile of the age distribution, and given the assumption above this doesn't vary much between, say, N=10^3 and N=10^6.)

It is sensitive in the obvious way to variation in that asymptotic mortality rate: if it's p then typical gaps between oldest-person deaths will be 1/p. (So, in particular, smallish errors in estimating p lead to smallish errors in estimating the time between oldest-person deaths.)


On the other hand, if mortality rate turns out to vary rapidly with age beyond some threshold (e.g., some malicious god kills everyone once they reach 120 or something) then the analysis above could be very wrong. In the extreme case where a malicious god kills everyone once they reach some easily-reachable age, the oldest-person death rate is just the birth rate times the probability of reaching the age in question.

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I fitted a power law to the mortality rate and extrapolated it. It appears to underestimate the acceleration of the mortality rate between 60 and 100 somewhat. However, an exponential fit (which is what I tried first) significantly overestimates the acceleration of the mortality rate between 60 and 100. If the true mortality rate increases faster than my extrapolation, it would increase the rate at which you get a new "oldest person in the world". Without better data I can't check this. Anyway, +1 :) –  Chris Taylor May 10 '13 at 14:09

I think that this question is best approached through careful modelling, rather than pure mathematics. Here's the approach I took. I don't claim that this is the perfect approach by any means, but it's a start.

Spoiler: My simulations give a rate of approximately once every 0.66 years, for a population of 7 billion people who share US mortality statistics.

First, I took the US mortality tables from the Center for Disease Control. They only go up to age 100, so I need to extrapolate beyond that. I fitted a power law to the hazard rate $h(a)$ which gives the probability of dying between age $a$ and $a+1$, getting

$$h(a) = 3.54 \times 10^{-15} \times a^{6.933}$$

I assume $h(a)=1$ in the case that my power law gives me a number above 1. This occurs at $a=122$, which seems realistic (the oldest person to ever live died at age 122).

I then simulated an evolving population until it converged on a stable distribution. I assume $N(a)$ people at age $A$, and a constant birth rate of $9\times 10^7$ people every year (chosen to give a stable popluation of 7 billion). The result is a reasonable-looking population pyramid:

enter image description here

Now that I have a stable population, I simulate again. For each age $a$, the number of people of age $a$ in year $t$ is the fraction of the population aged $a-1$ at time $t-1$ who don't die, i.e.

$$N(t,a) = (1-h(a-1)) \times N(t-1, a-1)$$

When appropriate I approximate the number of deaths with the normal distribution, but for small populations I use the binomial distribution. In the case that there are some deaths in the highest age bracket, I calculate the probability that the person who died was the oldest person in the world at that time, and record this as an event.

Taking the total number of events, and dividing by the number of years that I run the simulation for, gives an approximate rate. The punchline is that in my simulation, I see 15,234 events in 10,000 years, for an approximate rate of once in every 0.66 years.

Assuming a population of one billion people (the population of the developed world, to which the US mortality statistics are most likely to apply) we can see the following histogram, which gives the age of the oldest person in the world at the time they die. Comparing to the wikipedia page for oldest people it looks as though the numbers are too high by 1-2 years, but otherwise I'm surprised at how accurate this crude model is!

enter image description here

One final chart. This is how the number of deaths of the oldest living person each year varies as a function of the total population. Roughly, it seems to be linear in the logarithm of the population. I'd be interested to see a more rigorous mathematical treatment that can get this result out

enter image description here

Edit: I corrected a bug which was causing me to estimate the rate as too high. I was approximating the binomial $B(n,p)$ with a normal distribution $\sigma=np(1-p)$ rather than $\sigma^2=np(1-p)$.

Edit no. 2: It was pointed out in the comments that I had another bug, and I also realized that I wasn't ever checking for the possibility that more than 1 'oldest person' dies in a given year.

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Your model, with mortality rate accelerating rapidly at extreme ages, gives results rather close to my crude argument above. I can't help thinking it should yield rather more frequent oldest-person deaths than that. (And it appears that oldest-person deaths are quite a bit more frequent.) Is it possible that there's still some bug? –  Gareth McCaughan May 10 '13 at 14:06
    
@GarethMcCaughan It's possible! I'll put the code in a gist so you can check it (unfortunately you won't be able to run it as it uses some of my proprietary library of utilities... but you can at least check it for errors manually). –  Chris Taylor May 10 '13 at 14:11
    
@GarethMcCaughan My code is in a gist here: gist.github.com/chris-taylor/5554669 –  Chris Taylor May 10 '13 at 14:17
    
@Gugg Yeap, this mechanism is called HackerNews: news.ycombinator.com/item?id=5685579 –  Ivan May 10 '13 at 14:19
    
@gugg I deleted it to keep the answer simple, I could regenerate it and include it again. –  Chris Taylor May 10 '13 at 14:19

The Gerontology Research Group keeps records

I brought this up on the GRG and Louis Epstein posted the table "CHRONOLOGICAL OLDEST LIVING LISTED PERSONS (Since 1955)". I extracted the final column, death dates, and formatted it and extracted the intervals between the death dates of each person, reasoning that if the Oldest Person In The World who died in 1955 is succeeded by a person who died in 1956, that meant an observer would, in 1955, wait ~1 year for the new Oldest Person to die. The mean interval between deaths turns out to be 1.2 years, but the median wait turns out to be 0.65 years! This seems to be due in large part due to the astounding lifespan of Jeanne Calment, as you will see on the interval graph shortly:

deaths <- as.Date(c("1955-10-24","1959-06-25","1961-02-10","1964-12-30",
                    "1965-08-06","1966-01-10","1968-03-21","1968-06-16",
                    "1970-01-11","1973-02-27","1973-08-18","1973-10-31",
                    "1975-05-31","1976-11-16","1977-12-02","1978-04-25",
                    "1981-01-22","1981-03-09","1982-11-13","1983-10-13",
                    "1985-02-16","1986-10-21","1987-02-02","1987-12-27",
                    "1988-01-11","1997-08-04","1998-04-16","1999-12-30",
                    "2000-11-02","2001-06-06","2002-03-18","2003-10-31",
                    "2003-11-13","2004-05-29","2006-08-27","2006-12-11",
                    "2007-01-24","2007-01-28","2007-08-13","2008-11-26",
                    "2009-01-02","2009-09-11","2010-05-02","2010-11-04",
                    "2011-06-21","2012-12-04","2012-12-17"))

plot(deaths)

Plot of years of each date:

enter image description here

R> intervals <- NULL; for (i in 1:(length(deaths)-1)) { intervals[i] <- deaths[i+1] - deaths[i] }
R> intervals
 [1] 1340  596 1419  219  157  801   87  574 1143  172   74  577  535  381  144 1003   46  614  334
[20]  492  612  104  328   15 3493  255  623  308  216  285  592   13  198  820  106   44    4  197
[39]  471   37  252  233  186  229  532   13
R> summary(intervals)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
      4     147     270     454     588    3490
R> # the monstrous 3493 outlier is Jeanne Louise Calment:
R> # her predecessor Florence Knapp died in 1988, she in 1997

plot(intervals)

Graph the size of successive interval between the death of the previous Oldest Person and the replacement Oldest Person:

enter image description here

I leave it to you guys to compare the empirical observations with the mathematical models; what's interesting to me is that eyeballing it, there seems to be a sort of linear trend downwards where the Oldest Person is dying faster over time. This could be due to bad data collection (if the real Oldest Person is 110 but you only found some 105 piker, then your fake Oldest person will probably live longer than the real Oldest Person did) improving over time, or maybe a more interesting phenomenon where medicine is improving or the population is growing, and so there are many more centenarians, and so the Oldest Person is ever closer to the extreme of what is possible and naturally slips off the cliff that much quicker (from this perspective, Calment is even more of an extraordinary outlier).

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Thanks for this gwern. Love your site, by the way! –  Chris Taylor May 11 '13 at 10:39
    
You're welcome. (It's GRG's data, not mine, so thank them for collecting it; also, thanks to Gugg for editing in the images.) –  gwern May 11 '13 at 15:08
    
@gwern Many thanks! Doing some checks on the numbers you provided in the second grey block, I found a mean of 1.24 years and a median of 0.74 years. (Not your 0.65 years.) Taking out Calment (for no legitimate reason that I can think of, but still), brings those down to 1.06 and 0.70 years, resp. Did I do this correct? –  Glen The Udderboat May 11 '13 at 18:59
    
I might've messed up turning the median into years, yeah; I forget exactly what I did. –  gwern May 11 '13 at 23:33

Since all men are mortal it ALWAYS happens that the oldest person alive dies.

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Why the minus vote? My answer is perfectly true. It is peculiar that someone can disagree with this simple reasoning. –  Godot May 11 '13 at 15:33
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If you wanted to be clever, I suppose you should have answered "Exactly once". –  TMM May 11 '13 at 16:01
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This "answer" is based on a misunderstanding of the question, which is stated sufficiently clearly. Even assuming you read only the title of the question and not its body (which deserves an automatic -1), the question asks about how frequently the actual event of dying happens. It's not "When does it happen that the oldest person alive eventually dies?". For each person, the person dies only once; they're not dying every day, let alone "always". (Except to the extent that we're all dying a little every day, every minute.) –  ShreevatsaR May 11 '13 at 16:18

Super-simpler answer: "How often will the oldest living person die?" is just a paraphrase of "How long, typically, will the oldest person (on earth/in country/in any given population) live?" According to one source (http://life-span.findthedata.org/q/101/166/What-is-the-life-expectancy-of-a-100-year-old-man), life expectancy for a man is 2.07 years, for a woman 2.37 years, which is consistent with the 50%-per-year mortality rate mentioned earlier. It seems reasonable to assume that the oldest living person is "very old" in this sense. And since one person will stay the oldest living person until they die, questions of population size are irrelevant. Naturally, there will be variation from population to population, but since this is a probabalistic question anyway, "two and a quarter years" seems a reasonable answer, yes?

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It seems very reasonable. Right until you compare it to the data provided by @gwern. They average 1.24 years and only 7 of 46 are above 2.07 years. That seems to require some explanation beyond your answer. –  Glen The Udderboat May 12 '13 at 18:02
    
Completely wrong. You have answered the question 'What is the life expectancy of the oldest living person?' You should have answered the question 'What is the life expectancy of the oldest living person, at the time that the previous oldest person dies?' This is sensitive to population size, although only slightly. –  jwg Aug 1 '13 at 9:19

This hasn't happened in like 2000 years (http://en.wikipedia.org/wiki/Wandering_Jew) and since earth is only 6000 years old. So I would say not very often.

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Never, the oldest person alive is, by definition, alive, and thus can't be dead.

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3  
But they can die... –  Lord_Farin Jun 20 '13 at 18:26

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