Related papers: Halfspace depth for general measures: The ray basi…

We provide new theoretical results in the field of inverse regression methods for dimension reduction. Our approach is based on the study of some empirical processes that lie close to a certain dimension reduction subspace, called the…

We propose halfspace depth concepts for scatter, concentration and shape matrices. For scatter matrices, our concept is similar to those from Chen, Gao and Ren (2017) and Zhang (2002). Rather than focusing, as in these earlier works, on…

We present a criterion for the stochastic completeness of a submanifold in terms of its distance to a hypersurface in the ambient space. This relies in a suitable version of the Hessian comparison theorem. In the sequel we apply a…

The computation of the Tukey depth, also called halfspace depth, is very demanding, even in low dimensional spaces, because it requires the consideration of all possible one-dimensional projections. In this paper we propose a random depth…

In 1975 John Tukey proposed a multivariate median which is the 'deepest' point in a given data cloud in R^d. Later, in measuring the depth of an arbitrary point z with respect to the data, David Donoho and Miriam Gasko considered…

Data depth is a concept in multivariate statistics that measures the centrality of a point in a given data cloud in $\IR^d$. If the depth of a point can be represented as the minimum of the depths with respect to all one-dimensional…

The notion of data depth has long been in use to obtain robust location and scale estimates in a multivariate setting. The depth of an observation is a measure of its centrality, with respect to a data set or a distribution. The data depths…

The concept of statistical depth extends the notions of the median and quantiles to other statistical models. These procedures aim to formalize the idea of identifying deeply embedded fits to a model that are less influenced by…

Tukey's depth (or halfspace depth) is a widely used measure of centrality for multivariate data. However, exact computation of Tukey's depth is known to be a hard problem in high dimensions. As a remedy, randomized approximations of Tukey's…

We consider a generalization of the hyperplane problem to arbitrary measures in place of volume and to sections of lower dimensions. We prove this generalization for unconditional convex bodies and for duals of bodies with bounded volume…

We study generalizations of classical metric embedding results to the case of quasimetric spaces; that is, spaces that do not necessarily satisfy symmetry. Quasimetric spaces arise naturally from the shortest-path distances on directed…

As a measure for the centrality of a point in a set of multivariate data, statistical depth functions play important roles in multivariate analysis, because one may conveniently construct descriptive as well as inferential procedures…

A generalized divergence theorem is established allowing for domains with inner boundaries. The normal trace of a rough integrand is not a Radon measure; rather, the boundary integral is expressed via a surface functional continuous with…

This paper proposes a new approach to address the problem of unmeasured confounding in spatial designs. Spatial confounding occurs when some confounding variables are unobserved and not included in the model, leading to distorted…

We introduce a new intrinsic metric in subdomains of a metric space and give upper and lower bounds for it in terms of well-known metrics. We also prove distortion results for this metric under quasiregular maps.

Motivated by the problem of nonparametric inference in high level digital image analysis, we introduce a general extrinsic approach for data analysis on Hilbert manifolds with a focus on means of probability distributions on such sample…

The Subspace Theorem is a powerful tool in number theory. It has appeared in various forms and been adapted and improved over time. It's applications include diophantine approximation, results about integral points on algebraic curves and…

A data depth measures the centrality of a point with respect to an empirical distribution. Postulates are formulated, which a depth for functional data should satisfy, and a general approach is proposed to construct multivariate data depths…

We study empirical variants of the halfspace (Tukey) depth of a probability measure $\mu$, which are obtained by replacing $\mu$ with the corresponding weighted empirical measure. We prove analogues of the Marcinkiewicz--Zygmund strong law…

This is a Research and Instructional Development Project from the U. S. Naval Academy. In this monograph, the basic methods of nonstandard analysis for n-dimensional Euclidean spaces are presented. Specific rules are deveoped and these…