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March, 1990 On a Notion of Data Depth Based on Random Simplices
Regina Y. Liu
Ann. Statist. 18(1): 405-414 (March, 1990). DOI: 10.1214/aos/1176347507


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.


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Regina Y. Liu. "On a Notion of Data Depth Based on Random Simplices." Ann. Statist. 18 (1) 405 - 414, March, 1990.


Published: March, 1990
First available in Project Euclid: 12 April 2007

zbMATH: 0701.62063
MathSciNet: MR1041400
Digital Object Identifier: 10.1214/aos/1176347507

Primary: 62H05
Secondary: 60D05, 62H12

Rights: Copyright © 1990 Institute of Mathematical Statistics


Vol.18 • No. 1 • March, 1990
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