## DIMACS TR: 2003-28

##
Simplicial Depth: An Improved Definition, Analysis, and Efficiency for
the Finite Sample Case

### Authors: Michael A. Burr, Eynat Raflin and Diane L.

**
ABSTRACT
**

As proposed by Liu 1990 the simplicial depth of a point $x$ with respect
to a probability distribution $F$ on $R^d$ is the probability that $x$
belongs to a random simplex in $R^d$.
The simplicial depth of $x$ with respect to a data set $S$ in $R^d$ is
the fraction of the closed simplices given by $d+1$ of the data points
containing the point $x$.
We propose an alternative definition for simplicial depth which
continues to remain valid over a continuous probability field, but also
fixes some of the problems for the finite sample case, including those
discussed by Zuo and Serfling 2000. Additionally, we discuss the effect
of the revised definition on the efficiency of previously developed
algorithms and prove tight bounds on the value of the simplicial depth
based on the half-space depth.

Paper available at
ftp://dimacs.rutgers.edu/pub/dimacs/TechnicalReports/TechReports/2003/2003-28.ps.gz.

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