Partial Orders with Applications in Nonparametric Multivariate Analysis

Abstract

Many statisticians think that the obstacle to the development of multivariate nonparametric methods is the lack of natural total order in multidimensional space. In fact, Kendall (1938), in his rank correlation coefficient Tau, implicitly decomposes the order relation in two-dimensional space into two partial orders. Then, Kendall's Tau, written as a vector of two components, can be interpreted as the fractions of two partial orders contained in sample set. In this talk, the notion of partial orders will be extended to general k-dimensional space. Accordingly, Kendall's Tau, expressed in a vector form, can be extended for k variates. The population version of this generalized Tau is called partial-order probabilities (POP). Alternative to the traditional correlation matrix, POP can be used as association parameters for k variates. To demonstrate this role, POP will be used to construct a nonparametric test of independence for random vectors and a test of homogeneity in association between different samples. In addition, one can describe the relative location of two multivariate populations by partial orders, and hence it can be used to generalize the univariate Mann-Whitney test to multivariate case.