The Annals of Statistics

Ordering Directional Data: Concepts of Data Depth on Circles and Spheres

Regina Y. Liu and Kesar Singh

Full-text: Open access


Three notions of depth for directional data, angular simplicial depth (ASD), angular Tukey's depth (ATD) and arc distance depth (ADD), are developed and studied. The empirical versions of these depths give rise to center-outward rankings of angular data which may be regarded as extensions of the usual center-outward ranking on the line. Three medians derived from these depths are examined and compared. Applications in nonparametric classification and in implementing the bootstrap to construct confidence regions for directional parameters are briefly discussed.

Article information

Ann. Statist., Volume 20, Number 3 (1992), 1468-1484.

First available in Project Euclid: 12 April 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62H05: Characterization and structure theory
Secondary: 62H12: Estimation 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

Angular simplicial depth angular Tukey's depth arc distance depth median directional data center-outward ordering von Mises class of distributions


Liu, Regina Y.; Singh, Kesar. Ordering Directional Data: Concepts of Data Depth on Circles and Spheres. Ann. Statist. 20 (1992), no. 3, 1468--1484. doi:10.1214/aos/1176348779.

Export citation