On General Notions of Depth for Regression

@article{Zuo2018OnGN,
  title={On General Notions of Depth for Regression},
  author={Y. Zuo},
  journal={arXiv: Methodology},
  year={2018}
}
  • Y. Zuo
  • Published 2018
  • Mathematics
  • arXiv: Methodology
Depth notions in location have fascinated tremendous attention in the literature. In fact data depth and its applications remain one of the most active research topics in statistics in the last two decades. Most favored notions of depth in location include Tukey (1975) halfspace depth (HD), Liu (1990) simplicial depth, and projection depth (Stahel (1981) and Donoho (1982), Liu (1992), Zuo and Serfling (2000) (ZS00) and Zuo (2003)), among others. Depth notions in regression have also been… Expand
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References

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Asymptotics for the maximum regression depth estimator
Notions of depth in regression have been introduced and studied in the literature. Regression depth (RD) of Rousseeuw and Hubert (1999) (RH99), the most famous one, is a direct extension of TukeyExpand
Similarities between location depth and regression
The location depth of Tukey (1975) is a multivariate generalization of rank, and leads to a multivariate median known as the Tukey median. Recently, Rousseeuw and Hubert (1999a) introduced a notionExpand
Computing location depth and regression depth in higher dimensions
TLDR
An exact algorithm to compute the location depth in three dimensions in O(n2logn) time and an approximate algorithm that computes the depth of a regression fit in O3+mpn+mnlogn time are constructed. Expand
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This article introduces a halfspace depth in the location–scale model that is along the lines of the general theory given by Mizera, based on the idea of Rousseeuw and Hubert, and is complemented byExpand
Perspectives on Depth Functions on General Data Spaces , with Consideration of the Tukey , Projection , Spatial , “ Density ” , “ Local ” , and “ Contour ” Depths
General perspectives on the depth approach are developed in order to aid the interpretation of developments and findings to date and to provide viewpoints apropos to continuing development. TheseExpand
From Depth to Local Depth: A Focus on Centrality
Aiming at analyzing multimodal or nonconvexly supported distributions through data depth, we introduce a local extension of depth. Our construction is obtained by conditioning the distribution toExpand
General notions of statistical depth function
Statistical depth functions are being formulated ad hoc with increasing popularity in nonparametric inference for multivariate data. Here we introduce several general structures for depth functions,Expand
Data depths satisfying the projection property
Summary:Data depth is a concept that measures the centrality of a point in a given data cloud x1, x2,...,xn ∈ ℝ or in a multivariate distribution PX on ℝdd. Every depth defines a family of so–calledExpand
Projection-based depth functions and associated medians
order √ n uniform consistency. Depth regions and contours induced from projection-based depth functions are investigated. Structural properties of depth regions and contours and general continuityExpand
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 statisticalExpand
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