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

Abstract

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 $Z_1, \cdots, Z_N$ come from populations which have the joint density $\Pi^N_{i = 1} \frac{1}{\sigma\sqrt{2\pi}} \exp \big\lbrack - \frac{1}{2\sigma^2} (z_i - d_i\xi - \eta)^2 \big\rbrack,$ where $d_1, \cdots, d_N$ are given constants, not all equal, and $\xi/\sigma$ is sufficiently small. The test criterion was found to be $c_1(R) = \sum d_iEZ_{N, r_i}$, where $EZ_{Ni}$ is the expected value of the $i$th standard normal order statistic and $R = (r_1, \cdots, r_N)$ is the permutation of the ranks. The distribution of this statistic was shown to be asymptotically normal providing the known constants $d_1, \cdots, d_N$ satisfied Noether's condition [9]. The two-sample distribution is a special case, and the resultant statistic $c_1(R)$ is shown to be asymptotically normal. The approximation of the distribution of the $c_1(R)$ statistic to the distribution $C(1 - x^2)^{\frac{1}{2}N-2}, - 1 \leqq x \leqq 1$, 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 $c_1(R)$ 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 $c_1(R)$ for $N \leqq 10, p \leqq.10$.