(i) If X1,...,Xm and Y1,...,Yn are samples with continuous cumulative distribution functions F and G = h(F) respectively, and if h is differentiable, the distribution of the ranks S1 < ... < Sn of the Y ’s is given by P{S1 = s1,...,Sn = sn} = E 

h


U(s1)

...h


U(sn)



m+n m

(6.66)

where U(1) < ··· < U(m+n) is an ordered sample from the uniform distribution U(0, 1).

(ii) If in particular G = F k, where k is a positive integer, (6.66) reduces to P{S1 = s1,...,Sn = sn} (6.67)

= kn

m+n m

n j=1

Γ (sj + jk − j)

Γ (sj ) · Γ (sj+1)

Γ (sj+1 + jk − j)

.