Multivariate statistics plays a role of ever increasing importance in the modern era of information technology. Using the center-outward ranking induced by the notion of data depth, we describe several nonparametric tests of location and scale differences for multivariate distributions. The tests for location differences are derived from graphs in the so-called DD plots (depth vs. depth plots) and are implemented through the idea of permutation tests. The proposed test statistics are scale-standardized measures for the location difference and they can be carried out without estimating the scale or variance of the underlying distributions. The test for scale differences introduced in Liu and Singh (2003) is a natural multivariate rank test derived from the center-outward depth ranking and it extends the Wilcoxon rank-sum test to the testing of multivariate scale. We discuss the properties of these tests, and provide simulation results as well as a comparison study under normality. Finally, we apply the tests to compare airlines’ performances in the context of aviation safety evaluations.
"New Nonparametric Tests of Multivariate Locations and Scales Using Data Depth." Statist. Sci. 19 (4) 686 - 696, November 2004. https://doi.org/10.1214/088342304000000594