The Annals of Statistics

On a Notion of Data Depth Based on Random Simplices

Regina Y. Liu

Full-text: Open access

Abstract

For a distribution $F$ on $\mathbb{R}^p$ and a point $x$ in $\mathbb{R}^p$, the simplical depth $D(x)$ is introduced, which is the probability that the point $x$ is contained inside a random simplex whose vertices are $p + 1$ independent observations from $F$. Mathematically and heuristically it is argued that $D(x)$ indeed can be viewed as a measure of depth of the point $x$ with respect to $F$. An empirical version of $D(\cdot)$ gives rise to a natural ordering of the data points from the center outward. The ordering thus obtained leads to the introduction of multivariate generalizations of the univariate sample median and $L$-statistics. This generalized sample median and $L$-statistics are affine equivariant.

Article information

Source
Ann. Statist. Volume 18, Number 1 (1990), 405-414.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
http://projecteuclid.org/euclid.aos/1176347507

Digital Object Identifier
doi:10.1214/aos/1176347507

Mathematical Reviews number (MathSciNet)
MR1041400

Zentralblatt MATH identifier
0701.62063

JSTOR
links.jstor.org

Subjects
Primary: 62H05: Characterization and structure theory
Secondary: 62H12: Estimation 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

Keywords
Simplex simplicial depth multivariate median $L$-statistics angularly symmetric distributions location estimators consistency affine equivariance

Citation

Liu, Regina Y. On a Notion of Data Depth Based on Random Simplices. Ann. Statist. 18 (1990), no. 1, 405--414. doi:10.1214/aos/1176347507. http://projecteuclid.org/euclid.aos/1176347507.


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