On smoothness of Tukey depth contours. Issue 5 (2nd September 2016)

• Record Type: Journal Article
• Authors: Gijbels, Irène; Nagy, Stanislav
• Is Part Of: Statistics. Volume 50:Issue 5(2016)
• Page Start: 1075
• Page End: 1085
• Publication Date: 2016-09-02
• Subjects: data depth -- depth contours -- optimal halfspace -- Tukey depth -- smoothness -- quasi-concavity; Mathematical statistics -- Periodicals; 519.505
• DOI: 10.1080/02331888.2016.1145680
• Languages: English
• ISSNs: 0233-1888

Abstract

Abstract : The smoothness of Tukey depth contours is a regularity condition often encountered in asymptotic theory, among others. This condition ensures that the Tukey depth fully characterizes the underlying multivariate probability distribution. In this paper we demonstrate that this regularity condition is rarely satisfied. It is shown that even well-behaved probability distributions with symmetrical, smooth and (strictly) quasi-concave densities may have non-smooth Tukey depth contours, and that the smoothness behaviour of depth contours is fairly unpredictable.