Electronic Journal of Statistics

A new concept of quantiles for directional data and the angular Mahalanobis depth

Christophe Ley, Camille Sabbah, and Thomas Verdebout

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Abstract

In this paper, we introduce a new concept of quantiles and depth for directional (circular and spherical) data. In view of the similarities with the classical Mahalanobis depth for multivariate data, we call it the angular Mahalanobis depth. Our unique concept combines the advantages of both the depth and quantile settings: appealing depth-based geometric properties of the contours (convexity, nestedness, rotation-equivariance) and typical quantile-asymptotics, namely we establish a Bahadur-type representation and asymptotic normality (these results are corroborated by a Monte Carlo simulation study). We introduce new user-friendly statistical tools such as directional DD- and QQ-plots and a quantile-based goodness-of-fit test. We illustrate the power of our new procedures by analyzing a cosmic rays data set.

Article information

Source
Electron. J. Statist., Volume 8, Number 1 (2014), 795-816.

Dates
First available in Project Euclid: 16 June 2014

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1402927498

Digital Object Identifier
doi:10.1214/14-EJS904

Mathematical Reviews number (MathSciNet)
MR3217789

Zentralblatt MATH identifier
1349.62197

Keywords
Bahadur representation directional statistics DD- and QQ-plot Mahalanobis depth rotationally symmetric distributions

Citation

Ley, Christophe; Sabbah, Camille; Verdebout, Thomas. A new concept of quantiles for directional data and the angular Mahalanobis depth. Electron. J. Statist. 8 (2014), no. 1, 795--816. doi:10.1214/14-EJS904. https://projecteuclid.org/euclid.ejs/1402927498


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