Re: Dynamical Systems and Expansion-Contraction

>>From Osher Doctorow

It may look as though a hard problem is posed by Probable Correlation
in PI:

1) P(X<-->Y)(x, y) = F(x, y) + P(X > x, Y > y)

but statisticians do have a few tricks up their sleeves although not
always the ability to use them in unusual places.

By what is an incredible coincidence, statisticians of advanced
theoretical type have been studying Stochastic Orderings and Stochastic
Majorization, which roughly speaking examine which probability
distributions are "better" than others in the sense that for example:

2) F1(x, y) > = F2(x, y) for all x, y

where F1 and F2 are two bivariate cdfs for continuous random variables
X, Y. There are also orderings like:

3) FX1(x) > = FX2(x)

for all x, where FX1 and FX2 are two univariate cdfs (marginals).
Obviously, if P(X > x, Y > y) is constant, then P(X<-->Y) increases as
stochastic ordering or majorization increases from equation (1).

Stochastic orderings or majorization have a large variety of
modifications and analogs include ordering by expectations, variances,
etc.

There are also fascinating applications of stochastic orderings to
failure rates. For example, the Marshall and Olkin (1967)
multivariate exponential distribution has the MIFRA property, which
for a random nonnegative component random vector X = (X1, ..., Xn)
holds by definition iff:

4) E[h(X)] < = {E[h^a(X/a)]}^(1/a)

for every nonnegative nondecreasing function h and every a in (0, 1).
MIFRA marginals are also MIFRA, MIFRA distributions as a class of
distributions in R^n is closed under weak limits,(X, Y) is MIFRA is X
and Y are and X, Y are independent, and MIFRA class is closed under
convolutions. Also the Johnson and Kotz (1977) multivariate gamma
distribution has several MIFRA random variables associated with it.

Look up some of these topics as keywords on the internet, and I'll try
to continue soon.

Osher Doctorow

.

Loading