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TR20031
Simplicial Depth: An Improved Definition, Analysis, and Efficiency for the Finite Sample Case 

Authors:  Burr, Michael A.; Rafalin, Eynat; Souvaine, Diane L. 
Date:  January, 2003 
Pages:  17 
Download Formats:  [PDF] 
As proposed by Liu 1990 the simplicial depth of a point with respect to a probability distribution on is the probability that belongs to a random simplex in . The simplicial depth of with respect to a data set in is the fraction of the closed simplices given by of the data points containing the point . 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 halfspace depth. 
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