Sign Tests in Multidimension: Inference Based on the Geometry of the Data Cloud

Abstract

Multivariate sign tests attracted several statisticians in the past, and it is evident from recent nonparametric literature that they still continue to draw attention. One of the most important features of the univariate sign test is that it does not involve much technical assumptions or complicacy, and this makes it quite popular among statistics users. In this article we have come up with a new method for constructing multivariate sign tests that have reasonable statistical properties and can be used conveniently to solve one-sample location problems. Our principal strategy here is to make a wise utilization of certain geometric structures in the constellation of data points for making inference about the location of their distribution. As we proceed with the development of a fairly broad and general methodology, we indicate its relationship with previous work done by others and sometimes attempt to unify some of the earlier ideas. In particular, we pick up some well-known tests for uniform distribution of directional data and integrate them into the technology of multivariate sign tests to synthesize useful new procedures. Our procedures enjoy affine invariance and the distribution-free property for elliptically symmetric models. We report several interesting results that provide powerful insights into certain critical aspects of the problem. What is most appealing is the fundamental dependence of our approach on the basic geometry of the data cloud formed by the observations. In this article our only key to unlock the information contained in the data is the spatial arrangement of data points in a d-dimensional Euclidean space.

Key Words:

• Affine invariance
• Beran's family of tests
• Data-driven coordinates
• Degenerate U statistics
• Distribution-free tests
• Elliptically symmetric distributions