Some Rank Order Tests which are most Powerful Against Specific Parametric Alternatives

The most powerful rank order tests against specific parametric alternatives are derived. Following the methods of Hoeffding 4, we derive the most powerful rank order test of whether N observations come from the same but unknown population against the alternative that the observations Z₁, , ZN come from populations which have the joint density N₈ = ₁ 12 - 12² (zᵢ - dᵢ -) ², where d₁, , dN are given constants, not all equal, and / is sufficiently small. The test criterion was found to be c₁ (R) = dᵢEZ₍, ₑ㶁, where EZ₍₈ is the expected value of the ith standard normal order statistic and R = (r₁, , rN) is the permutation of the ranks. The distribution of this statistic was shown to be asymptotically normal providing the known constants d₁, , dN satisfied Noether's condition 9. The two-sample distribution is a special case, and the resultant statistic c₁ (R) is shown to be asymptotically normal. The approximation of the distribution of the c₁ (R) statistic to the distribution C (1 - x²) ^1{2N-2}, - 1 x 1, is investigated. This statistic is then compared to the existing Mann and Whitney U statistic. No method having been found for analytical evaluation of the power of this test, the power was examined experimentally. Tables are appended giving the exact distribution of the c₁ (R) statistic for all possible subsample sizes whose total size is less than or equal to 10 together with the corresponding Mann and Whitney U value. Table 2 gives critical values of c₁ (R) for N 10, p. 10.