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 ΠNi = 1 1σ√2π exp - 1/2σ² (zᵢ - dᵢξ - η)² , where d₁, ⋯, dN are given constants, not all equal, and ξ/σ is sufficiently small. The test criterion was found to be c₁(R) = ∑ dᵢEZN, rᵢ, where EZNi 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.