From Osher Doctorow
"Copula Theory" in modern mainstream mathematical
probability-statistics essentially makes use of the remarkable fact
that in the unit square or unit cube or unit higher dimensional object
of similar type, the following is relevant:
Theorem. The cumulative distribution function (cdf) of any random
variabl X, FX(x), with argument x replaced by a random variable X, so
that FX(X) is obtained, is itself a uniformly distributed random
variable on (0, 1) and therefore itself has the cdf:
1) cdf of FX(x) = x (true for any uniform distribution on (0, 1)
Professors Ferguson of UCLA and Kotz of Stanford have been among the
pioneers of this type of research. It is also very valuable for
computer simulation of any random variable.
Let's use this in combination with the previous postings of this
thread.
Gumbel's bivariate or joint exponential cdf discussed earlier is:
2) F(x,y) = 1 - exp(-x) - exp(-y) + exp[-(x + y + theta xy))
We've already derived the fourth term on the right hand side by PI, so
let's look the second and third terms -exp(-x) - exp(-y) which are of
course -FX(x) and -FY(y) respectively for the standard exponential
(marginal) distribution. But the same argument used in the first
posting of part 4 of this thread for the fourth term of (2) works for x
+ y, and from the above Theorem this is FX(x) + FY(y) in the unit
square, and so when we leave the unit square FX(x) = exp(-x), FY(y) =
exp(-y). So the second and third terms are explained.
Why do we replace x by x + y in the first posting of part 4? It is the
fundamental generalization of Probable Influence (PI) to generalize
with + or - rather than with multiplication or division or other
operations including composition in general. So to generalize to two
dimensions, x is replaced by x + y, although x - y could also have been
used. If x + x were used (or x - x), nothing would have been
accomplished non-trivially in two dimensions.
Osher Doctorow
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