Let Zi have a continuous cumulative distribution function Fi

(i = 1,...,N), and let G be the group of all transformations Z

i = f(Zi) such that f is continuous and strictly increasing.

(i) The transformation induced by f in the space of distributions is F

i =

Fi(f −1).

(ii) Two N-tuples of distributions (F1,...,FN ) and (F

1,...,F

N ) belong to the same orbit with respect to G¯ if and only if there exist continuous distribution functions h1,...,hN defined on (0,1) and strictly increasing continuous distribution functions F and F’ such that Fi = hi(F) and F

i = hi(F

).

[(i): P{f(Zi) ≤ y} = P{Zi ≤ f −1(y)} = Fi[f −1(y)].

(ii): If Fi = hi(F) and the F

i are on the same orbit, so that F

i = Fi(f −1), then F

i = hi(F

) with F
= F(f −1). Conversely, if Fi = hi(F), F

i = hi(F

), then F

i = Fi(f −1) with f = F
−1(F).]