Efficiency and Optimality Results for Tests Based on Weighted Rankings

Abstract

The Tukey–Quade method of weighted rankings for the analysis of complete blocks is considered. Assuming only that the within-block error components are exchangeable, the limiting distribution of a large class of test statistics based on this method is obtained under the null hypothesis and under nearby alternatives as the number of blocks increases to infinity. This makes it possible to derive the asymptotic efficiency of the allied tests relative to the maximin most powerful test based on block-location-free statistics. In particular, the Quade test is seen to be more efficient than the Friedman test when there are less than eight observations per block and the errors are assumed to be normally distributed. It is also possible to determine which within-block scores, which measure of block variability, and which between-block score-generating function maximize the asymptotic efficiency. It is seen that the class of test statistics based on the method of weighted rankings compares favorably with the class of test statistics based on the method of unweighted rankings. The intrinsically optimal asymptotic efficiencies of both classes are studied in detail when the errors are normally distributed and when they have the extreme value distribution.

Key Words:

• Distribution-free test
• Randomized blocks
• Method of n rankings
• Hájek-Šidák approach
• Maximin most powerful test