Two-Sample, Bivariate Hypothesis Testing Methods Based on Tukey's Depth

Abstract

The article considers methods for comparing bivariate distributions corresponding to two independent groups, all of which are based in part on Tukey's depth which provides one way of extending the notion of ranks to multivariate data. The methods add perspective on how groups differ and they address a technical issue that is typically ignored when using rank-based methods or robust measures of location. The first method uses measures of location as defined by Donoho and Gasko, and the other uses estimated depths of points directly. One version of the second approach represents a natural extension of the Wilcoxon-Mann-Whitney test to bivariate data. The other variation is designed to be sensitive to differences in scale. When comparing groups based on measures of location, it was found that a nonstandard bootstrap method is necessary to control the probability of a Type I error.