In article <[EMAIL PROTECTED]>,
Glen Barnett <[EMAIL PROTECTED]> wrote:
>Giuseppe Andrea Paleologo wrote:
>> I am dealing with a simple conjecture. Given two generic positive random
>> variables, is it always true that the sum of the quantiles (for a given
>> value p) is greater or equal than the quantile of the sum?
>> In other words, let X, Y be positive random variables with continuous
>> but arbitrary joint CDF F(x,y), and let Z = X + Y, with CDF Fz(z). Let
>> Fx(x) and Fy(y) are the marginal CDFs for X and Y respectively. Is it
>> true that
>> Fx^-1 (p) + Fy^-1 (p) >= Fz^-1(p)
>> with 0 < p < 1 ?
>> Any insight or counterexample is greatly appreciated. I am sure this is
>> proved in some textbook, but independently from that, I think this
>> should be doable via elementary methods...
The statement is false. It is easy to give a discrete
counterexample, and then to fudge it to make it
continuous. We can even have X and Y independent and
identically distributed.
So let X and Y be Bernoulli, with P(X=0) = .6. Then
if p = .5, which avoids questions of when one is found
in an interval, Fx^-1(p) = 0, Fy^-1(p) = 0, and
Fz^-1(p) = 1.
Now if one makes things continuous by adding iid
continuous positive random variables U and V,
independent of X and Y, such that P(U < .1) = .9,
and now write X for X+U and Y for Y+V, we have
Fx^-1(.5) < .1, but Fz^-1(.5) > 1.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
[EMAIL PROTECTED] Phone: (765)494-6054 FAX: (765)494-0558
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