Open Access
2020 Uniform convergence rates for the approximated halfspace and projection depth
Stanislav Nagy, Rainer Dyckerhoff, Pavlo Mozharovskyi
Electron. J. Statist. 14(2): 3939-3975 (2020). DOI: 10.1214/20-EJS1759


The computational complexity of some depths that satisfy the projection property, such as the halfspace depth or the projection depth, is known to be high, especially for data of higher dimensionality. In such scenarios, the exact depth is frequently approximated using a randomized approach: The data are projected into a finite number of directions uniformly distributed on the unit sphere, and the minimal depth of these univariate projections is used to approximate the true depth. We provide a theoretical background for this approximation procedure. Several uniform consistency results are established, and the corresponding uniform convergence rates are provided. For elliptically symmetric distributions and the halfspace depth it is shown that the obtained uniform convergence rates are sharp. In particular, guidelines for the choice of the number of random projections in order to achieve a given precision of the depths are stated.


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Stanislav Nagy. Rainer Dyckerhoff. Pavlo Mozharovskyi. "Uniform convergence rates for the approximated halfspace and projection depth." Electron. J. Statist. 14 (2) 3939 - 3975, 2020.


Received: 1 October 2019; Published: 2020
First available in Project Euclid: 22 October 2020

zbMATH: 07270282
MathSciNet: MR4165498
Digital Object Identifier: 10.1214/20-EJS1759

Primary: 62G20
Secondary: 62H12

Keywords: approximation , depth , halfspace depth , projection depth , Tukey depth

Vol.14 • No. 2 • 2020
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