On a notion of data depth based on random simplices. (English) Zbl 0701.62063

Liu, Regina Y.

Ann. Stat. 18, No. 1, 405-414 (1990).

Summary: For a distribution F on \({\mathbb{R}}^ p\) and a point x in \({\mathbb{R}}^ p\), the simplicial 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(\(\cdot)\) 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.

MSC:

62H05	Characterization and structure theory for multivariate probability distributions; copulas
62H12	Estimation in multivariate analysis
60D05	Geometric probability and stochastic geometry

Keywords:

new notion of data depth; angularly symmetric distributions; location estimators; consistency; affine equivariance; simplicial depth; random simplex; L-statistics; generalized sample median