# 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} }

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

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

- MathematicsStatistical 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

- Mathematics
- 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…

### 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

- MathematicsStatistics 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…

### Smooth depth contours characterize the underlying distribution

- MathematicsJ. Multivar. Anal.
- 2010

### Choosing Among Notions of Multivariate Depth Statistics

- MathematicsStatistical 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

- Geology
- 2019

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

- Mathematics
- 2013

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

- MathematicsStats
- 2020

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…

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