## The Annals of Statistics

### On a Notion of Data Depth Based on Random Simplices

Regina Y. Liu

#### 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

https://projecteuclid.org/euclid.aos/1176347507

Digital Object Identifier
doi:10.1214/aos/1176347507

Mathematical Reviews number (MathSciNet)
MR1041400

Zentralblatt MATH identifier
0701.62063

JSTOR