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.