Under the assumptions of the preceding problem, if Fi = hi(F), the distribution of the ranks T1,..., TN of Z1,..., ZN depends only on the hi , not on F. If the hi are differentiable, the distribution of the Ti is given by P{T1 = t1,..., TN = tn} =

E $

h 1

U(t1)

... h N

U(tN )

%

N! , (6.66)

where U(1) < ··· < U(N) is an ordered sample of size N from the uniform distribution U(0, 1). [The left-hand side of (6.66) is the probability that of the quantities F(Z1), . . . , F(ZN ), the ith one is the ti th smallest for i = 1,..., N. This is given by  ...  h 1(y1)... h N (yN ) dy integrated over the region in which yi is the ti th smallest of the y’s for i = 1,..., N. The proof is completed as in Problem 6.44.]