Multivariate Rank Functions and Related Depth and Quantile Functions

• Activity Number: 674
• Type: Topic Contributed
• Date/Time: Thursday, August 13, 2015 : 10:30 AM to 12:20 PM
• Sponsor: Section on Nonparametric Statistics
• Author(s): Robert Serfling* and Yunfei Wang
• Companies: The University of Texas at Dallas and The University of Texas at Dallas
• Keywords: Multivariate ; Rank functions ; Quantile functions ; Depth functions ; Nonparametric

Abstract

Diverse notions of multivariate rank functions have arisen. A unifying formulation as vectors in the unit ball is provided, whereby the inverse defines a quantile function and the magnitude an outlyingness function whose own inverse defines a depth function. New depth functions are found in this way and are discussed. The point where the sample rank function equals the zero vector defines a multivariate median, and evaluation at a hypothesized value provides a multivariate sign test. Certain rank functions yield the classical Hodges Sign Test and Blumen Sign Test. For a sample rank function defined relative to pairwise averages of the data, the point where it equals the zero vector defines a Hodges-Lehmann center, and evaluation at a hypothesized value provides a multivariate signed-rank test. These and certain generalized sign and signed-rank tests are discussed. Existing rank functions are compared with respect to invariance, robustness, efficiency, and computational burden. Open issues are discussed.