From Osher Doctorow
The bivariate joint logistic distribution turns out to have a positive
second mixed partial derivative of P(X<-->Y)(x,y) = F(x,y) + R(x,y) as
did previous distributions discussed here.
The best presentation of this distribution other than the original
paper by Gumbel is Johnson and Kotz (1972) (Volume 4 of their
comprehensive series on univariate and multivariate statistical
distributions and their applications which began in 1970 to my
recollection), page 291. They describe Gumbel's multivariate logistic
distribution, but the bivariate case is:
1) f(x, y) = 2(1 + exp(-x) + exp(-y))^(-3) exp(-x - y), x > 0, y > 0
It turns out again that Dxy[R(x, y)] = f(x, y) > 0 for x, y > 0.
A bivariate or multivariate distribution is generally named after its
marginals, so here "bivariate joint logistic" or just "bivariate
logistic" or "joint logistic" means that the marginals or univariate
pdfs are both logistic - in fact, they are standard logistic pdfs.
Osher Doctorow
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