Β-Skeleton depth functions and medians. Issue 20 (18th October 2018)

• Record Type: Journal Article
• Authors: Yang, Mengta; Modarres, Reza
• Is Part Of: Communications in statistics. Volume 47:Issue 20(2018)
• Page Start: 5127
• Page End: 5143
• Publication Date: 2018-10-18
• Subjects: Data depth -- β-skeleton -- multivariate median -- proximity graph; Mathematical statistics -- Periodicals; Mathematics; Statistics; 519.2
• DOI: 10.1080/03610926.2017.1386320
• Languages: English
• ISSNs: 0361-0926

Abstract

ABSTRACT: We define the β-skeleton depth based on the probability that a point is contained within the β-skeleton influence region of two i.i.d. random vectors. The proposed family of depth functions satisfies the four desirable properties of statistical depth function. We also define and examine the sample β-skeleton depth functions and show that they share well-behaved asymptotic properties, including uniform consistency and asymptotic normality. Finally, we explore the β-skeleton multidimensional medians as location estimators of the center of multivariate distributions, discuss its asymptotic properties, and study its breakdown point. A Monte Carlo study compares the β-skeleton medians with the random Tukey median and the sample mean.