Lens data depth and median

Zhenyu Liu and Reza Modarres

Journal of Nonparametric Statistics, 2011, vol. 23, issue 4, 1063-1074

Abstract: We define the lens depth (LD) function LD(t; F) of a vector t∈Rd with respect to a distribution function F to be the probability that t is contained in a random hyper-lens formed by the intersection of two closed balls centred at two i.i.d observations from F. We show that LD is a statistical depth function and explore its properties, including affine invariance, symmetry, maximality at the centre and monotonicity. We define the sample LD and investigate its uniform consistency, asymptotic normality and computational complexity in high-dimensional settings. We define the lens median (LM), a multivariate analogue of the univariate median, as the point where the LD is maximised. The sample LM is the vector that is covered by the most number of hyper-lenses formed between any two sample observations. We state its asymptotic consistency and normality and examine its breakdown point and relative efficiency. The sample LM is robust and efficient for estimating the centre of a unimodal distribution. A comparison of LD and LM to existing data depth functions and medians in terms of computational complexity, robustness, efficiency and breakdown point is presented.

Date: 2011
References: View complete reference list from CitEc
Citations: View citations in EconPapers (4)

Downloads: (external link)
http://hdl.handle.net/10.1080/10485252.2011.584621 (text/html)
Access to full text is restricted to subscribers.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:taf:gnstxx:v:23:y:2011:i:4:p:1063-1074

Ordering information: This journal article can be ordered from
http://www.tandfonline.com/pricing/journal/GNST20

DOI: 10.1080/10485252.2011.584621

Access Statistics for this article

Journal of Nonparametric Statistics is currently edited by Jun Shao

More articles in Journal of Nonparametric Statistics from Taylor & Francis Journals
Bibliographic data for series maintained by Chris Longhurst ().