Multivariate Nonparametric Several-Sample Tests

Abstract

Let $(X^{(1)}_{j1}, \cdots, X^{(p)}_{j1}), \cdots, (X^{(1)}_{jn_j}, \cdots, X^{(p)}_{jn_j})$ be random samples of size $n_j$ from the $c$ continuous $p$-variate distribution functions $F_j(\mathbf{x}) = F(\mathbf{x} - \mathbf{\theta}_j)$ where $\mathbf{x} = (x_1, \cdots, x_p), \mathbf{\theta}_j = (\theta_{ij}, \cdots, \theta_{pj})$ and $j = 1, \cdots, c$. This paper is concerned with the problem of testing the hypothesis $H : \mathbf{\theta}_1 = \mathbf{\theta}_2 = \cdots = \mathbf{\theta}_c$ against the alternative that all $\mathbf{\theta}_j$ are not equal on the basis of the above $c$ samples. When we especially consider the asymptotic efficiency of test, the following alternative $K$ is adopted, $K : \mathbf{\theta}_j = \mathbf{\theta} + \mathbf{\nu}_j/N^{\frac{1}{2}},\quad\mathbf{\nu}_j = (\nu_{ij}, \cdots, \nu_{pj}),\quad j = 1, \cdots, c.$ We shall develop in this paper some test procedures for the hypothesis $H$ which are originated from the paper of Chernoff-Savage [2]. When $c = 2$, that is multivariate two-sample tests, Sugiura [7] has proposed a test statistic of Wilcoxon type. On the other hand, Puri [5] has derived univariate several-sample tests in general type including the test of Kruskal-Wallis [4]. It will be shown that the latter are corresponding to the case $p = 1$ in this paper and the former is a particular one among the case $c = 2$ of our tests.