Help with efficient double sum of max (X_i, Y_i) (X & Y vectors)

Jeff,

you can do

sum1: \sum_i\sum_j max(X_i,X_j)
sum2: \sum_i\sum_j max(Y_i,Y_j)

sum(x * (2 * rank(x) - 1))

sum3: \sum_i\sum_j max(X_i,Y_j)

sum(outer(x, y, pmax))

Probably, the latter can be speeded up even more...
Z