The hypothesis of randomness.

7 Let Z1,...,ZN be independently distributed with distributions F1,...,FN , and let Ti denote the rank of Zi among the Z’s For testing the hypothesis of randomness F1 = ··· = FN against the alternatives K of an upward trend, namely that Zi is stochastically increasing with i, consider the rejection regions iti > C (6.70)
and iE(V(ti)) > C, (6.71)
where V(1) < ··· < V(N) is an ordered sample from a standard normal distribution and where ti is the value taken on by Ti.
(i) The second of these tests is most powerful among rank tests against the normal alternatives F = N(γ + iδ, σ2) for sufficiently small δ.
(ii) Determine alternatives against which the first test is a most powerful rank test.
(iii) Both tests are unbiased against the alternatives of an upward trend; so is any rank test φ satisfying φ(z1,...,zN ) ≤ φ(z

1,...,z

N ) for any two points for which i < j, zi < zj implies z

i < z

j for all i and j.
[(iii): Apply