Halfspace depth for general measures: the ray basis theorem and its consequences

@article{Laketa2021HalfspaceDF,
  title={Halfspace depth for general measures: the ray basis theorem and its consequences},
  author={Petra Laketa and Stanislav Nagy},
  journal={Statistical Papers},
  year={2021}
}
The halfspace depth is a prominent tool of nonparametric multivariate analysis. The upper level sets of the depth, termed the trimmed regions of a measure, serve as a natural generalization of the quantiles and inter-quantile regions to higher-dimensional spaces. The smallest non-empty trimmed region, coined the halfspace median of a measure, generalizes the median. We focus on the (inverse) ray basis theorem for the halfspace depth, a crucial theoretical result that characterizes the halfspace… 

References

SHOWING 1-10 OF 38 REFERENCES
The depth function of a population distribution
Abstract. Tukey (1975) introduced the notion of halfspace depth in a data analytic context, as a multivariate analog of rank relative to a finite data set. Here we focus on the depth function of an
Fast Computation of Tukey Trimmed Regions and Median in Dimension p > 2
TLDR
Two novel algorithms to compute a Tukey κ-trimmed region are constructed, a naïve one and a more sophisticated one that is much faster than known algorithms, and a strict bound on the number of facets of aTukey region is derived.
Convergence of depths and depth-trimmed regions
Depth is a concept that measures the `centrality' of a point in a given data cloud or in a given probability distribution. Every depth defines a family of so-called trimmed regions. For statistical
Monge-Kantorovich Depth, Quantiles, Ranks and Signs
We propose new concepts of statistical depth, multivariate quantiles,ranks and signs, based on canonical transportation maps between a distributionof interest on IRd and a reference distribution on
Halfplane trimming for bivariate distributions
Let [mu] be a probability measure on R2 and let u [set membership, variant] (0, 1). A bivariate u-trimmed region D(u), defined as the intersection of all halfplanes whose [mu]-probability measure is
Concentration of the empirical level sets of Tukey’s halfspace depth
Tukey’s halfspace depth has attracted much interest in data analysis, because it is a natural way of measuring the notion of depth relative to a cloud of points or, more generally, to a probability
Halfspace depth and floating body
Little known relations of the renown concept of the halfspace depth for multivariate data with notions from convex and affine geometry are discussed. Halfspace depth may be regarded as a measure of
Continuity of Halfspace Depth Contours and Maximum Depth Estimators: Diagnostics of Depth-Related Methods
Continuity of procedures based on the halfspace (Tukey) depth (location and regression setting) is investigated in the framework of continuity concepts from set-valued analysis. Investigated
Illumination Depth
Abstract The concept of illumination bodies studied in convex geometry is used to amend the halfspace depth for multivariate data. The proposed notion of illumination enables finer resolution of the
...
...