The Annals of Statistics

On a Notion of Data Depth Based on Random Simplices

Regina Y. Liu

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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

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

First available in Project Euclid: 12 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


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

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


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.

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