Properties of Statistical Depth with Respect to Compact Convex Random Sets: The Tukey Depth

Functional classification and the random Tukey depth. Practical issues

Depths are used to attempt to order the points of a multidimensional or infinite dimensional set from the “center of the set” to the “outer of it”. There are few definitions of depth which are valid in the functional case. One of them is the so-called random Tukey depth, which is based on some randomly chosen one-dimensional projections and thus varies (randomly) from computation to computation...

ABCDepth: efficient algorithm for Tukey depth

We present a new algorithm for Tukey (halfspace) depth level sets and its implementation. Given d-dimensional data set for any d ≥ 2, the algorithm is based on representation of level sets as intersections of balls in R, and can be easily adapted to related depths (Type D, Zuo and Serfling (Ann. Stat. 28 (2000), 461–482)). The algorithm complexity is O(dn + n log n) where n is the data set size...

Tukey Depth-based Multivariate Trimmed Means

We investigate the asymptotic behavior of two types of Tukey depth-based multivariate trimmed means. Sufficient conditions for asymptotic normality of these location estimators are given. Two approaches to trimming are distinguished and central limit theorems are derived for each one. Asymptotic normality is proved using Hadamard differentiability of the location functionals. In the one-dimensi...

Some Properties of Certain Subclasses of Close-to-Convex and Quasi-convex Functions with Respect to 2k-Symmetric Conjugate Points

Ray-Shooting Depth: Computing Statistical Data Depth of Point Sets in the Plane

Over the past several decades, many combinatorial measures have been devised for capturing the statistical data depth of a set of n points in R. These include Tukey depth [15], Oja depth [12], Simplicial depth [10] and several others. Recently Fox et al. [7] have defined the Ray-Shooting depth of a point set, and given a topological proof for the existence of points with high Ray-Shooting depth...