From Osher Doctorow
A key important indication of Probable Causation/Influence (PI) is
Positive Statistical Quadrant Dependence, for for short Positive
(Statistical) Dependence, which can be formulated as:
1) F(x, y) > FX(x)FY(y), F(x, y) the bivariate cdf, FX, FY marginals
where cdf = cumulative distribution function and the marginals are the
univeriate cdfs of X and Y respectively.
The corresponding condition for (Statistical) Independence is:
2) F(x, y) = FX(x)FY(y)
for all x, y in the relevant domain or range.
N. L. Johnson and either J. S. Kotz or Balakrishnan (1971 or 1972), in
their comprehensive volumes on univariate and multivariate
distributions, outline the proof that (page 251) for bivariate extreme
value distribution we have:
3) F_infinity(x, y) > = F_infinityX(x)F_infinityY(y) where
F_infinity(x, y) is the limiting joint distribution of X1, ..., Xn and
Y1, ..., Yn are n approaches infinity.
This is important because in general the marginals will not
necessarily be independent, so PI is quite clearly indicated there.
Furthermore, Negative Dependence is undesirable as argued by me in
earlier postings, because it would mean roughly that random variables
X and Y "go in opposite directions".
Note carefully that F(x, y) is the joint cdf of X_max = max(X1, ...,
Xn) and Y_max = max(Y1, ..., Yn) where (Xi, Yi) are independent pairs
of continuous random variables identically distirbuted with joint cdf
F(x, y) where X1, ..., Xn are iid (independent identically
distributed) continuous random variables. F_infinity(x, y) is the
limiting distribution of F(x, y) as n approaches infinity.
Osher Doctorow
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