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On a notion of simplicial depth.

For a distribution F on R(p) and a point x in R(p) the simplicial depth D(x), which is the probability that x be inside a random simplex whose vertices are p + 1 independent observations from F, is introduced. D(x) can be viewed as a measure of depth of the point x relative to F, and its empirical version gives rise to a natural ordering of the data points from the center outward. This ordering provides an approach for detecting outliers in a multivariate data cloud and leads to the introduction of affine equivariant multivariate generalizations of the univariate sample median and L-statistics. This sample median is shown to be consistent for the center of any angularly symmetric distribution.