In the preceding problem let Ui j = 1 if (j − i)(Z j − Zi) > 0, and

= 0 otherwise.

(i) The test statistic i Ti , can be expressed in terms of the U’s through the relation



N i=1 i Ti = 

i< j

(j − i)Ui j +

N(N + 1)(N + 2)

6 .

(ii) The smallest number of steps [in the sense of Problem 6.42(ii)] by which

(Z1,..., ZN ) can be transformed into the ordered sample (Z(1),..., Z(N)) is

[N(N − 1)/2] − U, where U = i< j Ui j . This suggests U > C as another rejection region for the preceding problem.
[(i): Let Vi j = 1 or 0 as Zi ≤ Zi or Zi > Z j . Then Tj = N i=1 Vi j , and Vi j = Ui j or 1 − Ui j as i < j or i ≥ j. Expressing N j=1 j Tj = N j=1 j N i=1 Vi j in terms of the U’s and using the fact that Ui j = Uji , the result follows by a simple calculation.]