A high-dimensional spatial rank test for two-sample location problems

Abstract

In high-dimensional situations, the traditional multivariate sign- or rank-based procedures for the two-sample location testing problems are ineffective, since in the construction of the test statistics, the scatter matrix to be inverted is singular. To solve this problem, many high-dimensional spatial sign or rank tests have been proposed, some of which are very efficient. However, most of these existing tests no longer work in very high dimensional situations, which only allows the dimension of variables to be the square of the sample sizes at most, hence are restrictive for practical applications. On this ground, a new high-dimensional spatial rank test is proposed in this paper, which is invariant under scalar transformations, maintains the efficiency advantage of spatial-rank-based testing methods, and could even allow the dimension to grow almost exponentially with the sample sizes. The theoretical results of the proposed test are established, followed by some convincing numerical results and two real data analyses.