# Continuity of Halfspace Depth Contours and Maximum Depth Estimators: Diagnostics of Depth-Related Methods

@article{Mizera2002ContinuityOH,
title={Continuity of Halfspace Depth Contours and Maximum Depth Estimators: Diagnostics of Depth-Related Methods},
author={Ivan Mizera and Milos Volauf},
journal={Journal of Multivariate Analysis},
year={2002},
volume={83},
pages={365-388}
}
• Published 1 November 2002
• Mathematics
• Journal of Multivariate Analysis
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 procedures are depth contours (upper level sets) and maximum depth estimators. Continuity is studied both as the pointwise continuity of data-analytic functions, and the weak continuity of statistical functionals--the latter having relevance for qualitative robustness. After a real-data example, some…
25 Citations

## Figures from this paper

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

• Mathematics
Statistical Papers
• 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

### M ay 2 02 1 HALFSPACE DEPTH FOR GENERAL MEASURES : THE RAY BASIS THEOREM AND ITS CONSEQUENCES

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

### Location–Scale Depth

• Computer Science
• 2004
A halfspace depth in the location–scale model is introduced that is along the lines of the general theory given by Mizera, based on the idea of Rousseeuw and Hubert, and is complemented by a new likelihood-based principle for designing criterial functions.

### Halfspace depth and floating body

• Mathematics
Statistics Surveys
• 2019
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

### Choosing Among Notions of Multivariate Depth Statistics

• Mathematics
Statistical Science
• 2022
Classical multivariate statistics measures the outlyingness of a point by its Mahalanobis distance from the mean, which is based on the mean and the covariance matrix of the data. A depth function is

### Multivariate Functional Halfspace Depth

• Mathematics
• 2012
This article defines and studies a depth for multivariate functional data. By the multivariate nature and by including a weight function, it acknowledges important characteristics of functional data,

### Algorithmic and geometric aspects of data depth with focus on $\beta$-skeleton depth

This thesis discusses three depth functions: two well-known depth functions halfspace depth and simplicial depth, and one recently defined depth function named as β-skeleton depth, β ≥ 1, which is equivalent to the previously defined spherical depth and lens depth.

### Depth for Vector-Valued Functions

Data depth is a nonparametric statistical tool applicable to multi- dimensional observations. In the contribution a depth suitable for vector-valued functional (innite-dimensio nal) data of one

### Depth Induced Regression Medians and Uniqueness

The notion of median in one dimension is a foundational element in nonparametric statistics. It has been extended to multi-dimensional cases both in location and in regression via notions of data

## References

SHOWING 1-10 OF 21 REFERENCES

### Asymptotic distributions of the maximal depth estimators for regression and multivariate location

• Mathematics
• 1999
We derive the asymptotic distribution of the maximal depth regression estimator recently proposed in Rousseeuw and Hubert. The estimator is obtained by maximizing a projection-based depth and the

### Convergence of depth contours for multivariate datasets

• Mathematics, Computer Science
• 1997
This work considers contour constructions based on a notion of data depth and proves a uniform contour convergence theorem under verifiable conditions on the depth measure, and applications to several existing depth measures are considered.

### On depth and deep points: a calculus

For a general definition of depth in data analysis a differential-like calculus is constructed in which the location case (the framework of Tukey's median) plays a fundamental role similar to that of

### Halfplane trimming for bivariate distributions

• Mathematics
• 1994
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

### Regression depth. Commentaries. Rejoinder

• Mathematics
• 1999
In this article we introduce a notion of depth in the regression setting. It provides the rank of any line (plane), rather than ranks of observations or residuals. In simple regression we can compute

### The Bagplot: A Bivariate Boxplot

• Mathematics
• 1999
Abstract We propose the bagplot, a bivariate generalization of the univariate boxplot. The key notion is the half space location depth of a point relative to a bivariate dataset, which extends the

### A Cautionary Note on the Method of Least Median Squares

• Mathematics
• 1992
Abstract This article describes and illustrates a local instability that may arise when using the method of least median squares (LMS) to fit models to data. This idea is contrary to the generally

### Sign Tests in Multidimension: Inference Based on the Geometry of the Data Cloud

• Mathematics, Computer Science
• 1993
A new method for constructing multivariate sign tests that have reasonable statistical properties and can be used conveniently to solve one-sample location problems is come up.

### Robust Statistics

The classical books on this subject are Hampel et al. (1986); Huber (1981), with somewhat simpler (but partial) introductions by Rousseeuw & Leroy (1987); Staudte & Sheather (1990). The dates reflect

### Robustness of Deepest Regression

• Mathematics
• 2000
In this paper we investigate the robustness properties of the deepest regression, a method for linear regression introduced by Rousseeuw and Hubert [6]. We show that the deepest regression functional