Beta-Skeletons Depth Functions and Medians

• Activity Number: 131
• Type: Contributed
• Date/Time: Monday, August 4, 2014 : 8:30 AM to 10:20 AM
• Sponsor: Section on Nonparametric Statistics
• Author(s): Mengta Yang*+ and Reza Modarres
• Companies: and George Washington University
• Keywords: Data Depth ; Beta-skeleton ; Proximity graph ; Multivariate median

Abstract

The beta-skeleton is a family of proximity graphs with applications in many scientific fields. We define the beta-skeleton depth with respect to a distribution function (DF) F to be the probability that a point is contained within the beta-skeleton influence region between two i.i.d. random vectors with DF F. We show that the beta-skeleton depth is a family of statistical depth functions that are monotonic, affine invariant, maximized at the center and vanishing at infinity. We define and examine the sample beta-skeleton depth function and show that it has desirable asymptotic properties, including uniform consistency and asymptotic normality. We also consider estimating the center of several multivariate distributions using the beta-skeleton multidimensional median, discuss its asymptotic properties and study its breakdown point. We compare the beta-skeleton median with the random Tukey median and the sample mean using a Monte Carlo study and note that its performance is competitive.