Re: Dynamical Systems and Expansion-Contraction

To compare Gumbel's bivariate exponential and Gumbel's bivariate
logistic cdfs, respectively:

1) F(x, y) = 1 - exp(-x) - exp(-y) + exp{-(x + y + axy)}
2) F1(x, y) = (1 + exp(-x) + exp(-y))^(-1)

we can use a few "tricks". Notice that in (1) we have x and y positive
and a in [0, 1], so F(x, y) has the form P(A-->B) = 1 + y - x with x
replaced by exp(-x) + exp(-y) and y replaced by exp{-(x + y + axy)}.
We know that we can make P(A-->B) = 1 + y - x arbitrarily close to 0 by
making x arbitrarily close to 1 and y arbitrarily close to 0 (why?),
under the condition that y < = x, and if we consider the ratio:

3) F(x, y)/F1(x, y)

the negative exponent -1 of F1 becomes a positive exponent 1 so that
the main parentheses term of F1 becomes greater than 1.

Translated into replacing x by exp(-x) + exp(-y) and replacing y by
exp{-(x + y + axy)}, we need to be able to choose exp(-x) + exp(-y)
near 1 and the second exponential near 0. If x and y are near 0.70,
then exp(-x) + exp(-y) is near 1, and if a is near 1 then the second
exponential is near 0.21 which is considerably closer to 0, and 1 times
0.21 is 0.21 < < 1 so the ratio in (3) is considerably less than 1 and
F(x, y) < F1(x, y). Note that a table of positive and negative
exponentials helps to visualize what is happening. In terms of
intervals, "near" above is in terms of "intervals near".

Osher Doctorow

.

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