A RANK-SUM TEST OF WHETHER TWO MULTIVARIATE SAMPLES WERE DRAWN FROM THE SAME POPULATION: PRELIMINARY REPORT

摘要

A discrete or qualitative variable g(x) defined on a finite Bet of H points {x,}, i=l ...N generates a partition into k disjoint subsets of n1...nk points, respectively, nl+...+nk=N, with each subset representing an equivalence class, xi = Xj iff g(xi) = g(xj). Under a statistical null hypothesis such a partition is random and each of the Hi/ ni possible partitions is assigned equal probability. Alternative hypotheses to be considered here are charac¬terized by some form of segregation where the distance d(Xi,Xj) between two points tends to be smaller for equivalent points than for non-equivalent points. A nonparametric measure of segregation is the proportion of times like neighbors are nearer than unlike neighbors, where each of the N points .is considered to have N-l neighbors. The mean (= r) and variance of this statistic are derived under the null hypothesis.nIntended extensions include the asymptotic theory of random partitions end the development of measures of attraction, repulsion, and buffering between equivalence classes. The hope is to provide answers to such questions as: if g(xi) < g(xi) < g(xk) then does the xj class tend to fall in between the x, and xk classes.