Linear Nonparametric Tests Against Restricted Alternatives: The Simple-Tree Order and The Simple Order

Abstract

The problem of testing homogeneity of distributions against some ordered alternatives is considered without making specific parametric model assumptions. The alternatives of interest include the simple-tree order and the simple order. The approach is based on a reformulation of the testing problems, regarding them as being composed of a finite number of two-sample sub-testing problems. The Mann-Whitney-Wilcoxon test is used, as an example, for the component problems, the main issue being how to combine dependent test statistics into a single efficient overall test. Keeping the practitioner’s needs in mind, a linear function of the individual test statistics is considered and the Abelson-Tukey-Schaafsma-Smid principle (of minimizing “the maximum shortcoming”) is applied to derive the optimal (minimax) coefficients in the linear combination. The advantages of using optimal weights are illustrated by making ARE comparisons with the Jonckheere-Terpstra-Tryon-Hettmansperger test. The restriction to the class of linear combinations is attractive from the practitioner’s viewpoint but the theoretical statistician will argue that such linear combinations leading to “somewhere” most powerful tests makes them questionable from an overall point of view. Hence, alternative approaches are discussed, at least to some extent. The main purpose of the paper, however, is to assist the practitioner who has decided to use a linear combination but worries about the weights to be chosen. It is argued that the restriction by linearity is less questionable for the simple order than for the simple-tree order.