Death of the oldest person
Robert Hill
Happy New Year and long life to all readers.
I recently saw a list of notable people who died in 1997.
They included the world's oldest person (Jeanne Calment, France, 122),
the oldest person in Britain (also female, of course), and the oldest
man in Britain.
This set me wondering on the question: in a country of known total
population, how often would one expect the event "death of the oldest
person in the country" to occur? What mathematics would one use
to estimate this? What plausible assumptions would one have to
make in order to arrive at a ball-park estimate (in the style of a
physicist) if one did not have access to very detailed demographic data?
For a start we can say that, for this problem at least, men are irrelevant.
The oldest person will essentially always be a woman. This means we can also
ignore any tendency for large numbers of men in some birth cohorts to die
young in wars. Let the total female population of the country be N.
Assume that conditions are typical of a modern developed country with
only slowly changing demographic parameters.
Where would one go from there?
--
Robert Hill
University Computing Service, Leeds University, England
"Though all my wares be trash, the heart is true."
- John Dowland, Fine Knacks for Ladies (1600)
.
man. As for the problem, I would surmise without rigorous statisical
analysis there would be absolutely no way to come reasonably close. In
fact meteorology which does use powerful computer simulations, satellite
data, and much more can't even predict the weather for a month in
advance with any astounding accuracy.
Pierre Abbat
> This set me wondering on the question: in a country of known total
> population, how often would one expect the event "death of the oldest
> person in the country" to occur? What mathematics would one use
> to estimate this? What plausible assumptions would one have to
> make in order to arrive at a ball-park estimate (in the style of a
> physicist) if one did not have access to very detailed demographic data?
It depends on the shape of the tail of the mortality curve. My father once
showed me an equation to estimate the mortality curve, but I forget it. If
the mortality curve were a brick wall, the event would happen as often as
birth; this is obviously not the case.
phma
John Shonder
Robert Hill wrote in message <1998Jan5.1...@leeds.ac.uk>...
(...)
>This set me wondering on the question: in a country of known total
>population, how often would one expect the event "death of the oldest
>person in the country" to occur? What mathematics would one use
>to estimate this? What plausible assumptions would one have to
>make in order to arrive at a ball-park estimate (in the style of a
>physicist) if one did not have access to very detailed demographic data?
>
>For a start we can say that, for this problem at least, men are irrelevant.
>The oldest person will essentially always be a woman. This means we can
also
>ignore any tendency for large numbers of men in some birth cohorts to die
>young in wars. Let the total female population of the country be N.
>Assume that conditions are typical of a modern developed country with
>only slowly changing demographic parameters.
>
>Where would one go from there?
>
Off the top of my head, I'd guess that the death of the oldest person in the
world occurs in almost every year: the life expectancy of a very old person
is likely to be very short. But if you really want an answer to the
question, numerical simulation is probably the best way to go. It wouldn't
be too hard to do. You would need some demographic data, a mortality table,
and some assumptions about the birth rate.
I don't know about other countries, but the U.S. Census Bureau publishes a
wealth of data, and a lot of it is available on-line (see www.census.gov ).
In the most recent "Statistical Abstract of the United States" (available
on-line at that site) there is a mortality table listing expected deaths per
1000 alive for people between birth and age 85. Using this data I found that
the "mean service life" of a human being in the U.S. is about 81 years. In
other words, by the time you're 81, half of your peers (people who were born
in the same year as you) are already dead. But an even more depressing fact
is that by the time you're 38, half of your life is already over!
Have a nice day.
John Shonder
Oak Ridge National Laboratory
Colin Rosenthal
>Off the top of my head, I'd guess that the death of the oldest person in the
>world occurs in almost every year: the life expectancy of a very old person
>is likely to be very short.
True enough, but the world's oldest person is generally someone who
has developed a very good talent for not dying. I wonder if there are
any reliable statistics for, say, the probability per unit time of
a 115 year old person dying? In any case, this particular old lady
had been the oldest person in the world for a number of years, and
I seem to remember her male Japanese predecessor also held the title
for a number of years.
But if you really want an answer to the
>question, numerical simulation is probably the best way to go. It wouldn't
>be too hard to do. You would need some demographic data, a mortality table,
>and some assumptions about the birth rate.
--
Colin Rosenthal
High Altitude Observatory
Boulder, Colorado
rose...@hao.ucar.edu
Robert Israel
|> This set me wondering on the question: in a country of known total
|> population, how often would one expect the event "death of the oldest
|> person in the country" to occur? What mathematics would one use
|> to estimate this? What plausible assumptions would one have to
|> make in order to arrive at a ball-park estimate (in the style of a
|> physicist) if one did not have access to very detailed demographic data?
Suppose births take place as a Poisson process with constant intensity b,
and the lifetime of each individual has distribution F (i.e. F(x) is the
probability of death by age x). Let A(t) be the age of the oldest person
alive at time t. Let H(y) = int_y^infinity dx (1-F(x)). In particular
H(0) is the life expectancy at birth.
Then it can be shown that
P(A(t) <= y) = exp(-b H(y))
(which is the probability that everyone born before time t-y is dead at
time t). Note that there is a positive, but presumably very small probability
exp(-b H(0)) that nobody is alive at time t.
Given that the oldest person alive at time t has age y,
the probability that this person survives to time t+x is (1-F(y+x))/(1-F(y)).
On the other hand, if nobody is alive at time t we have to wait an expected
time 1/b for someone to be born and then H(0) for that person to die.
Thus the expected time from now until the next death of an oldest person
is
exp(-b H(0))(1/b + H(0))
+ int_0^infinity dy (exp(-b H(y))' int_0^infinity dx (1-F(y+x))/(1-F(y))
= exp(-b H(0))(1/b + H(0))
+ int_0^infinity dy int_0^infinity dx b (1 - F(x+y)) exp(-b H(y))
= exp(-b H(0))(1/b + H(0)) + b int_0^infinity dy H(y) exp(-b H(y))
Robert Israel isr...@math.ubc.ca
Department of Mathematics (604) 822-3629
University of British Columbia fax 822-6074
Vancouver, BC, Canada V6T 1Z2
John Shonder
Colin Rosenthal wrote in message <68tplu$djj$1...@ncar.ucar.edu>...
>On Tue, 6 Jan 1998 11:05:04 -0500,
>John Shonder <shon...@ornl.gov> wrote:
>
>>Off the top of my head, I'd guess that the death of the oldest person in
the
>>world occurs in almost every year: the life expectancy of a very old
person
>>is likely to be very short.
>
>True enough, but the world's oldest person is generally someone who
>has developed a very good talent for not dying.
I disagree. The French woman who died last year at the age of 122 just
happened to be the last person born in 1876 to die. Somebody had to go last,
and it turned out to be her.
Colin Rosenthal
On Tue, 6 Jan 1998 15:06:56 -0500,
>
>Colin Rosenthal wrote in message <68tplu$djj$1...@ncar.ucar.edu>...
>>On Tue, 6 Jan 1998 11:05:04 -0500,
>>John Shonder <shon...@ornl.gov> wrote:
>>
>>>Off the top of my head, I'd guess that the death of the oldest person in
>the
>>>world occurs in almost every year: the life expectancy of a very old
>person
>>>is likely to be very short.
>>
>>True enough, but the world's oldest person is generally someone who
>>has developed a very good talent for not dying.
>
>I disagree. The French woman who died last year at the age of 122 just
>happened to be the last person born in 1876 to die. Somebody had to go last,
>and it turned out to be her.
I assume you have a sufficient database on death-rates among 120+ year-olds
to justify that statement :-)
Now actually I seem to remember hearing that for very old people the probability
of death per unit time is approximately constant, in which case the
prob. of the oldest person dying per unit time is just equal to that
number.
John Shonder
Colin Rosenthal wrote in message <68u577$soc$1...@ncar.ucar.edu>...
>Now actually I seem to remember hearing that for very old people the
probability
>of death per unit time is approximately constant, in which case the
>prob. of the oldest person dying per unit time is just equal to that
>number.
Not to belabor this, but... let p(t) be the probabilty that a person of age
t (in years) will survive for another year. The census bureau publishes this
data, but unfortunately it only goes up to age 80. Given this limitation, a
plot of ln(ln(1/p)) vs. t above age 50 falls on a line so straight (r^2 =
0.999) that I suspect this is the model they used to develop their mortality
table.
Robert Israel
In article <68u15g$ffd$1...@nntp.ucs.ubc.ca>, isr...@math.ubc.ca (Robert Israel) writes:
|> In article <1998Jan5.1...@leeds.ac.uk>, ec...@sun.leeds.ac.uk (Robert Hill) writes:
|> |> This set me wondering on the question: in a country of known total
|> |> population, how often would one expect the event "death of the oldest
|> |> person in the country" to occur? What mathematics would one use
|> |> to estimate this? What plausible assumptions would one have to
|> |> make in order to arrive at a ball-park estimate (in the style of a
|> |> physicist) if one did not have access to very detailed demographic data?
|> Suppose births take place as a Poisson process with constant intensity b,
|> and the lifetime of each individual has distribution F (i.e. F(x) is the
|> probability of death by age x). Let A(t) be the age of the oldest person
|> alive at time t. Let H(y) = int_y^infinity dx (1-F(x)). In particular
|> H(0) is the life expectancy at birth.
|> Then it can be shown that
|> P(A(t) <= y) = exp(-b H(y))
|> Thus the expected time from now until the next death of an oldest person
|> is
...
|> exp(-b H(0))(1/b + H(0)) + b int_0^infinity dy H(y) exp(-b H(y))
I didn't quite answer the question that was asked, if by "how often" you
mean the expected interval from one death-of-oldest-person to the
next. Note that by symmetry the answer I gave is equal to the expected
time from the last death-of-oldest-person to now, or half the expected
length of the interval containing "now", but because of the "inspection
paradox" the current interval tends to be longer than a typical interval.
Assume that the lifetime distribution F has a density f = F'.
Given that the age of the currently oldest person is y > 0, the probability
of a death-of-oldest person between now and now+h is f(y)/(1-F(y)) h + O(h^2).
On the other hand, if there is nobody currently alive the probability is O(h^2)
(because it requires both a birth and a death to occur). So the (unconditional)
probability of a death-of-oldest-person in this time interval is
h int_0^infinity dy f(y)/(1-F(y)) (exp(-b H(y))' + O(h^2)
= h b int_0^infinity dy f(y) exp(-b H(y)) + O(h^2)
The expected number of such deaths in a given time interval of length T is then
T b int_0^infinity dy f(y) exp(-b H(y))
and thus the expected interval should be
1/(b int_0^infinity dy f(y) exp(-b H(y)))
I should be careful here: the deaths-of-oldest-person are not a Markov process,
so you can't just use the Elementary Renewal Theorem, but I expect that for any
"reasonable" lifetime distribution the process has good enough ergodic
properties that this will work.
Note also that the expected number N of individuals alive at any given time
is related to the birth intensity b and the life expectancy H(0) by N = b H(0).
Rose Baker
Robert Israel wrote in message <68unmd$3or$1...@nntp.ucs.ubc.ca>...
>The expected number of such deaths in a given time interval of length T is
then
> T b int_0^infinity dy f(y) exp(-b H(y))
>and thus the expected interval should be
> 1/(b int_0^infinity dy f(y) exp(-b H(y)))
>I should be careful here: the deaths-of-oldest-person are not a Markov
process,
>so you can't just use the Elementary Renewal Theorem, but I expect that for
any
>"reasonable" lifetime distribution the process has good enough ergodic
>properties that this will work.
>
>Note also that the expected number N of individuals alive at any given time
>is related to the birth intensity b and the life expectancy H(0) by N = b
H(0).
>
I agree with this later mailing. Robert has said everything I would have
said if I had been faster off the mark! I'd just add that he has assumed a
steady-state population, and that it is possible to redo the calculation
with age distribution in the population not derived from the survival
distribution for individuals. These latter were published in 1991 and are
effectively Gompertz curves (exponentially increasing hazard of death) in
the UK. It would be possible to crank out an approximation to the integral,
using the method of steepest descents. To get the thing more accurate, we
would need to include the higher male mortality.
Pity (for this purpose) that lifespan is not exponentially distributed, when
the problem becomes much simpler---everyone has an equal probability of
dying in any time interval, and the gap between deaths of the oldest is (I
think from doing it in my head) just the average lifespan.
All this would make a nice student project!
Rose
onghappy
what is the oldest pple on the world?
120?150?189?
Three chioce
Pierre Abbat <ph...@pop.trellis.net> wrote in article
<01bd1a5f$a8bfcf80$4250...@phma.trellis.net>...
> > This set me wondering on the question: in a country of known total
> > population, how often would one expect the event "death of the oldest
> > person in the country" to occur? What mathematics would one use
> > to estimate this? What plausible assumptions would one have to
> > make in order to arrive at a ball-park estimate (in the style of a
> > physicist) if one did not have access to very detailed demographic
data?
>