The Annals of Mathematical Statistics

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

Milton E. Terry

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 $N \leqq 10, p \leqq.10$.

Article information

Source
Ann. Math. Statist., Volume 23, Number 3 (1952), 346-366.

Dates
First available in Project Euclid: 28 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177729381

Digital Object Identifier
doi:10.1214/aoms/1177729381

Mathematical Reviews number (MathSciNet)
MR49532

Zentralblatt MATH identifier
0048.36702

JSTOR