The Annals of Statistics

General notions of statistical depth function

Robert Serfling and Yijun Zuo

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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, classify many existing examples as special cases, and establish results on the possession, or lack thereof, of four key properties desirable for depth functions in general. Roughly speaking, these properties may be described as: affine invariance, maximality at center, monotonicity relative to deepest point, and vanishing at infinity. This provides a more systematic basis for selection of a depth function. In particular, from these and other considerations it is found that the halfspace depth behaves very well overall in comparison with various competitors.

Article information

Ann. Statist., Volume 28, Number 2 (2000), 461-482.

First available in Project Euclid: 15 March 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H05: Characterization and structure theory
Secondary: 62G20

Statistical depth functions halfspace depth simplicial depth multivariate symmetry


Zuo, Yijun; Serfling, Robert. General notions of statistical depth function. Ann. Statist. 28 (2000), no. 2, 461--482. doi:10.1214/aos/1016218226.

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