On a Notion of Data Depth Based on Random Simplices

For a distribution F on Rᵖ and a point x in Rᵖ, 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 () 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.