Nonparametric multivariate multisample tests of the location problem and multivariate regression based on directions of data

Randles' one sample multivariate sign test based on interdirections is extended to two sample and multisample tests. A heuristic argument suggests that when samples are drawn from elliptically symmetric distributions all of these tests share the same asymptotic relative efficiency with respect to conventional parametric multivariate one, two and multisample tests, respectively, under normality.; Next new nonparametric multivariate two sample and multisample tests are developed based on directions of differences between paired data, each member of a pair being from different samples. Then their asymptotic relative efficiencies with respect to Hotelling's {dollar}Tsp2{dollar} and Wilks' {dollar}Lambda{dollar} are calculated. The relationship of our statistics to Chaudhuri's location estimates based on U-statistics is discussed. Numerically evaluated asymptotic relative efficiencies and Monte Carlo simulation studies for the various distributions show that our tests are more powerful than Hotelling's {dollar}Tsp2{dollar} and Wilks' {dollar}Lambda{dollar} when samples are from heavy tailed distributions. They are also competitive among other nonparametric tests when samples are from light tailed distributions. We show that their power and level breakdown points are as good as those of the univariate sign test and Wilcoxon rank sum test.; Finally we apply this method to extend Kendall's {dollar}tau{dollar} to a multivariate setting and develop nonparametric multiple and multivariate regression techniques based on directions of data. Univariate and multivariate analysis of variance are considered as special cases of the regression problem with a fixed and balanced design matrix. As the dimension of the space where the explanatory variables take their values increases, the numerically evaluated asymptotic efficiency increases for each studied distribution. For independent random variables with values in a space of fixed dimension, its asymptotic relative efficiencies coincide with those of our location tests.