A Class of Non-Parametric Tests for Homogeneity Against Ordered Alternatives

Abstract

In this paper, the $c$-sample location problem with ordered or restricted alternatives is considered. Linear combinations of Chernoff-Savage type two-sample statistics computed among the $c(c - 1)/2$ pairs of samples are proposed as test statistics. It is shown that for each linear combination of two-sample statistics there is another linear combination, using only the $c - 1$ two-sample statistics based on adjacent samples as determined by the alternative, which has the same Pitman efficacy. If the ordered alternative is restricted further by specifying the relative spacings in the alternative, then the weighting coefficients can be chosen to maximize the Pitman efficacy over the class of linear combinations. It is also shown that the statistics proposed by Jonkheere [4] and Puri [9] have maximum Pitman efficacy when the alternative specifies equal spacings.