From Osher Doctorow

One of the most fascinating families of probability distributions is
the Frechet extremal distribution by Frechet (1951, 1958) which has the
advantage of being a comprehensive family of bivariate uniform
distributions (contains the product of the marginals or the
statistically independent case as a member of the family, etc.). Its
joint cdf is:

1) F(x,y) = [a^2(1 - a)/a]max(0, x + y - 1) + [a^2(1 + a)/2]min(x, y) +
(1 - a^2)xy, a constant (parameter) in [-1, 1]

where the marginals FX(x) = x, FY(y) = y for the uniform distribution
on [0, 1]. It turns out that:

Lemma. Dxy[F(x,y)] = 1 - a^2 for F(x,y) Frechet extremal

Proof. The case x + y - 1 0 and x y will be proven here; the others
are similar and are omitted for brevity. Under these conditions, the
max function above is 0, and the min function above is y, so we get:

2) F(x,y) = [a^2(1 + a)/2]y + (1 - a^2)xy

and the first partial derivative Dx(F(x,y)) with respect to x is (1 -
a^2)y, and the second Dxy(F(x,y)) is (1 - a^2). Q.E.D.

Corollary. Dxy[F(x,y)] 0 iff a^2 1 and Dxy[F(x,y)] increases
toward 1 as a^2 decreases toward 0, where a is the linear correlation
coefficient of X and Y.

Proof. Dxy[F(x,y)] 0 is direct from the Lemma. Similarly for
Dxy[F(x,y)] increasing toward 1 as a^2 decreases toward 0. To prove
that a is the correlation coefficient is a bit more difficult, but is
left as an exercise. Notice that when a = 0, then F(x,y) = xy which is
the uncorrelated and in fact independent case as we would expect for a
being the correlation coefficient, and when a = 1 then F(x,y) = min(x,
y) which is known to be the maximum correlation cdf as again would be
expected for a being the correlation coefficient. Q.E.D.

We aren't quite through in terms of the acceleration of the universe if
the distribution is Frechet extremal, since we still have to find
Dxy[R(x,y)]. However, since Dxy[F(x,y)] = f(x,y) which is the joint
probability density function (pdf), and R(x,y) is the integral from x
to infinity of the integral from y to infinity of f, which are both
truncated at 1, we just have to evaluate:

3) I1[I2(1 - rho^2)dy]dx

where I2...dy is the integral from y to 1 and I1...dx is the integral
from x to 1. We get for (3) the result 1 - rho^2 again, and therefo

4) Dxy[P(X--Y)(x,y) = Dxy[F(x,y)] + Dxy[R(x,y)] = 2(1 - rho^2)

and the universe accelerates for linear correlation squared rho^2 0
and the acceleration increases as rho^2 -- 0 as expected from the
previous examples.

So as the random variables X and Y become more and more linearly
uncorrelated so to speak, the acceleration of the universe increases
more and more in the positive range.

Osher Doctorow

From Osher Doctorow

An excellent presentation of the comprehensive properties of the
Frechet extremal distribution is in Luc Devroye (1986) Non-Uniform
Random Variable Generation, Springer-Verlag: N.Y. It is also
downloadable online thanks to the author from Devroye's site that can
be reached by just typing the keywords "Frechet extremal" or
http://cgm.cs.mcgill.ca/~luc/rnbookindex.html, etc.

Osher Doctorow

From Osher Doctorow

Hoeffding (1940) and Whitt (1976) showed that maximal positive and
negative correlation are obtained for the Frechet extremal distribution
functions. Hoeffding's work in fact has given rise to the name
Frechet-Hoeffding for the joint cdf and/or for related inequalities.
See "Modelling dependence with copulas and applications to risk
management," Paul Embrechts, Filip Lindskog, and Alexander McNeil
(Dept. Math. ETHZ Zurich) Sept. 10 2001,
www.math.ethz.ch/~baltes/ftp/copchapter.pdf, for more on the
Frechet-Hoeffding bounds for joint cdfs including various partial
orderings to which they give rise. Kotz has a paper on this too, which
I'll try to locate. Harry Joe's 1997 volume (U. British Columbia)
which I've mentioned before has whole chapters on various Frechet
distributions.

Osher Doctorow

From Osher Doctorow

Any joint bivariate cdf F(x,y) can be expressed as:

1) F(x,y) = C(FX(x), FY(y)), all real x, y

where C is a unique function in (1) called a "copula" introduced by A.
Sklar (1959) which maps the unit square [0, 1] X [0, 1] to the unit
interval [0, 1] and such that C(x, 0) = 0 = C(0, x) and C(x,1) = C(1,x)
= x whenever x is in [0, 1] and also C(x2,y2) - C(x2,y1) + C(x1,y1) -
C(x1,y2) \ 0 when 0 = x1 = x2 = 1 and 0 = y1 = y2 = 1.
See P. Mikusinski, H. Sherwood, and M. D. Taylor (Dept. Math, U. of
Central Florida, Orlando, Florida) in Mosler and Scarsini (Editors)
1991, "Probabilistic interpretations of copulas and their convex sums"
(this is the title of their paper, not of the book by Mosler and
Scarsini).

It turns out that (as Frechet proved in 1951):

2) max(0, FX(x) + FY(y) - 1) = F(x,y) = min(FX(x), FY(y))

for all real x, y. These are the Frechet or Frechet-Hoeffding bounds.

Osher Doctorow

I have never really thought about this problem, because the answer is so
obvious. The birth of a new group of stars is an explosion. These explosions
are like a pedigree, and every new generation has a higher initial velocity
compared to the first one.

Henry Haapalainen

Henry Haapalainen wrote:
I have never really thought about this problem, because the answer is so
obvious. The birth of a new group of stars is an explosion. These explosions
are like a pedigree, and every new generation has a higher initial velocity
compared to the first one.


"higher initial velocity" with respect to what?

"Sam Wormley" kirjoitti viestissä
news:%OFZe.365865$x96.145176@attbi_s72...
Henry Haapalainen wrote:
I have never really thought about this problem, because the answer is so
obvious. The birth of a new group of stars is an explosion. These
explosions
are like a pedigree, and every new generation has a higher initial
velocity
compared to the first one.


"higher initial velocity" with respect to what?

I am sorry that English is much more difficult to me than physics. Is this
better: ...a higher initial escaping velocity from the start point.

Henry Haapalainen

Henry Haapalainen wrote:
"Sam Wormley" kirjoitti viestissä
news:%OFZe.365865$x96.145176@attbi_s72...

Henry Haapalainen wrote:

I have never really thought about this problem, because the answer is so
obvious. The birth of a new group of stars is an explosion. These explosions
are like a pedigree, and every new generation has a higher initial velocity
compared to the first one.


"higher initial velocity" with respect to what?


I am sorry that English is much more difficult to me than physics. Is this
better: ...a higher initial escaping velocity from the start point.


I don't believe the is any evidence that that is the case.